homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk...
Transcript of homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk...
![Page 1: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/1.jpg)
Quantum functions and the Morita theory ofquantum graph isomorphisms
David Reutter
University of Oxford
Combining Viewpoints in Quantum TheoryICMS, Heriot-Watt University
March 20, 2018
David Reutter Quantum graph isomorphisms March 20, 2018 1 / 23
![Page 2: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/2.jpg)
Overview
This talk is based on joint work with Ben Musto and Dominic Verdon:A compositional approach to quantum functions
The Morita theory of quantum graph isomorphisms
quantuminformation
compact quantumgroups
category theory &higher algebra
quantum graph isomorphisms and their role in pseudo-telepathy[Atserias, Mancinska, Roberson, Samal, Severini, Varvitsiotis, 2017]
quantum automorphism groups of graphs[Banica, Bichon and others, 1999–2018]
Seems to fit into a much broaded framework of ’finite quantum set theory’.Part 1: Getting startedPart 2: Quantum functionsPart 3: The Morita theory of quantum graph isomorphisms
David Reutter Quantum graph isomorphisms March 20, 2018 2 / 23
![Page 3: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/3.jpg)
Overview
This talk is based on joint work with Ben Musto and Dominic Verdon:A compositional approach to quantum functions
The Morita theory of quantum graph isomorphisms
quantuminformation
compact quantumgroups
category theory &higher algebra
quantum graph isomorphisms and their role in pseudo-telepathy[Atserias, Mancinska, Roberson, Samal, Severini, Varvitsiotis, 2017]
quantum automorphism groups of graphs[Banica, Bichon and others, 1999–2018]
Seems to fit into a much broaded framework of ’finite quantum set theory’.Part 1: Getting startedPart 2: Quantum functionsPart 3: The Morita theory of quantum graph isomorphisms
David Reutter Quantum graph isomorphisms March 20, 2018 2 / 23
![Page 4: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/4.jpg)
Overview
This talk is based on joint work with Ben Musto and Dominic Verdon:A compositional approach to quantum functions
The Morita theory of quantum graph isomorphisms
quantuminformation
compact quantumgroups
category theory &higher algebra
quantum graph isomorphisms and their role in pseudo-telepathy[Atserias, Mancinska, Roberson, Samal, Severini, Varvitsiotis, 2017]
quantum automorphism groups of graphs[Banica, Bichon and others, 1999–2018]
Seems to fit into a much broaded framework of ’finite quantum set theory’.Part 1: Getting startedPart 2: Quantum functionsPart 3: The Morita theory of quantum graph isomorphisms
David Reutter Quantum graph isomorphisms March 20, 2018 2 / 23
![Page 5: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/5.jpg)
Overview
This talk is based on joint work with Ben Musto and Dominic Verdon:A compositional approach to quantum functions
The Morita theory of quantum graph isomorphisms
quantuminformation
compact quantumgroups
category theory &higher algebra
quantum graph isomorphisms and their role in pseudo-telepathy[Atserias, Mancinska, Roberson, Samal, Severini, Varvitsiotis, 2017]
quantum automorphism groups of graphs[Banica, Bichon and others, 1999–2018]
Seems to fit into a much broaded framework of ’finite quantum set theory’.Part 1: Getting startedPart 2: Quantum functionsPart 3: The Morita theory of quantum graph isomorphisms
David Reutter Quantum graph isomorphisms March 20, 2018 2 / 23
![Page 6: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/6.jpg)
Overview
This talk is based on joint work with Ben Musto and Dominic Verdon:A compositional approach to quantum functions
The Morita theory of quantum graph isomorphisms
quantuminformation
compact quantumgroups
category theory &higher algebra
quantum graph isomorphisms and their role in pseudo-telepathy[Atserias, Mancinska, Roberson, Samal, Severini, Varvitsiotis, 2017]
quantum automorphism groups of graphs[Banica, Bichon and others, 1999–2018]
Seems to fit into a much broaded framework of ’finite quantum set theory’.Part 1: Getting startedPart 2: Quantum functionsPart 3: The Morita theory of quantum graph isomorphisms
David Reutter Quantum graph isomorphisms March 20, 2018 2 / 23
![Page 7: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/7.jpg)
Overview
This talk is based on joint work with Ben Musto and Dominic Verdon:A compositional approach to quantum functions
The Morita theory of quantum graph isomorphisms
quantuminformation
compact quantumgroups
category theory &higher algebra
quantum graph isomorphisms and their role in pseudo-telepathy[Atserias, Mancinska, Roberson, Samal, Severini, Varvitsiotis, 2017]
quantum automorphism groups of graphs[Banica, Bichon and others, 1999–2018]
Seems to fit into a much broaded framework of ’finite quantum set theory’.
Part 1: Getting startedPart 2: Quantum functionsPart 3: The Morita theory of quantum graph isomorphisms
David Reutter Quantum graph isomorphisms March 20, 2018 2 / 23
![Page 8: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/8.jpg)
Overview
This talk is based on joint work with Ben Musto and Dominic Verdon:A compositional approach to quantum functions
The Morita theory of quantum graph isomorphisms
quantuminformation
compact quantumgroups
category theory &higher algebra
quantum graph isomorphisms and their role in pseudo-telepathy[Atserias, Mancinska, Roberson, Samal, Severini, Varvitsiotis, 2017]
quantum automorphism groups of graphs[Banica, Bichon and others, 1999–2018]
Seems to fit into a much broaded framework of ’finite quantum set theory’.Part 1: Getting startedPart 2: Quantum functionsPart 3: The Morita theory of quantum graph isomorphisms
David Reutter Quantum graph isomorphisms March 20, 2018 2 / 23
![Page 9: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/9.jpg)
Part 1Getting started
David Reutter Quantum graph isomorphisms March 20, 2018 3 / 23
![Page 10: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/10.jpg)
Quantum Functions
fSet ' (cfdC∗Alg)op ↪→ (fdC∗Alg)op ' fqSet
{finite setsfunctions
}
commutative
f.d. C ∗-algebras∗-homomorphisms
op {
f.d. C ∗-algebras∗-homomorphisms
}op
set of functions fSet(A,B)
quantum set of maps fqSet(A,B)
How to ‘quantize’ morphisms?
Sketch-Definition:Quantum set of quantum functions ! internal hom [A,B] in C∗Algop
X ,Y finite sets. [X ,Y ] = universal C ∗-algebrawith generators {pxy}x∈X ,y∈Y and relations:
p∗xy = pxy = p2xy pxypxy ′ = δy ,y ′pxy∑y∈Y
pxy = 1
Variant: [1]Group of bijections Sn ! internal ‘automorphism group’ in C∗Algop
internal hom is infinite-dimensional operator algebras
compositional nature of quantum functions obscured
physical meaning and applications?
David Reutter Quantum graph isomorphisms March 20, 2018 4 / 23
![Page 11: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/11.jpg)
Quantum Functions
fSet
' (cfdC∗Alg)op ↪→ (fdC∗Alg)op ' fqSet{
finite setsfunctions
} commutative
f.d. C ∗-algebras∗-homomorphisms
op {
f.d. C ∗-algebras∗-homomorphisms
}op
set of functions fSet(A,B)
quantum set of maps fqSet(A,B)
How to ‘quantize’ morphisms?
Sketch-Definition:Quantum set of quantum functions ! internal hom [A,B] in C∗Algop
X ,Y finite sets. [X ,Y ] = universal C ∗-algebrawith generators {pxy}x∈X ,y∈Y and relations:
p∗xy = pxy = p2xy pxypxy ′ = δy ,y ′pxy∑y∈Y
pxy = 1
Variant: [1]Group of bijections Sn ! internal ‘automorphism group’ in C∗Algop
internal hom is infinite-dimensional operator algebras
compositional nature of quantum functions obscured
physical meaning and applications?
David Reutter Quantum graph isomorphisms March 20, 2018 4 / 23
![Page 12: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/12.jpg)
Quantum Functions
fSet '
(cfdC∗Alg)op ↪→ (fdC∗Alg)op ' fqSet{
finite setsfunctions
}
commutative
f.d. C ∗-algebras∗-homomorphisms
op
{f.d. C ∗-algebras∗-homomorphisms
}op
set of functions fSet(A,B)
quantum set of maps fqSet(A,B)
How to ‘quantize’ morphisms?
Sketch-Definition:Quantum set of quantum functions ! internal hom [A,B] in C∗Algop
X ,Y finite sets. [X ,Y ] = universal C ∗-algebrawith generators {pxy}x∈X ,y∈Y and relations:
p∗xy = pxy = p2xy pxypxy ′ = δy ,y ′pxy∑y∈Y
pxy = 1
Variant: [1]Group of bijections Sn ! internal ‘automorphism group’ in C∗Algop
internal hom is infinite-dimensional operator algebras
compositional nature of quantum functions obscured
physical meaning and applications?
David Reutter Quantum graph isomorphisms March 20, 2018 4 / 23
![Page 13: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/13.jpg)
Quantum Functions
fSet ' (cfdC∗Alg)op
↪→ (fdC∗Alg)op ' fqSet{
finite setsfunctions
} commutative
f.d. C ∗-algebras∗-homomorphisms
op {
f.d. C ∗-algebras∗-homomorphisms
}op
set of functions fSet(A,B)
quantum set of maps fqSet(A,B)
How to ‘quantize’ morphisms?
Sketch-Definition:Quantum set of quantum functions ! internal hom [A,B] in C∗Algop
X ,Y finite sets. [X ,Y ] = universal C ∗-algebrawith generators {pxy}x∈X ,y∈Y and relations:
p∗xy = pxy = p2xy pxypxy ′ = δy ,y ′pxy∑y∈Y
pxy = 1
Variant: [1]Group of bijections Sn ! internal ‘automorphism group’ in C∗Algop
internal hom is infinite-dimensional operator algebras
compositional nature of quantum functions obscured
physical meaning and applications?
David Reutter Quantum graph isomorphisms March 20, 2018 4 / 23
![Page 14: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/14.jpg)
Quantum Functions
fSet ' (cfdC∗Alg)op ↪→
(fdC∗Alg)op ' fqSet{
finite setsfunctions
} commutative
f.d. C ∗-algebras∗-homomorphisms
op
{f.d. C ∗-algebras∗-homomorphisms
}op
set of functions fSet(A,B)
quantum set of maps fqSet(A,B)
How to ‘quantize’ morphisms?
Sketch-Definition:Quantum set of quantum functions ! internal hom [A,B] in C∗Algop
X ,Y finite sets. [X ,Y ] = universal C ∗-algebrawith generators {pxy}x∈X ,y∈Y and relations:
p∗xy = pxy = p2xy pxypxy ′ = δy ,y ′pxy∑y∈Y
pxy = 1
Variant: [1]Group of bijections Sn ! internal ‘automorphism group’ in C∗Algop
internal hom is infinite-dimensional operator algebras
compositional nature of quantum functions obscured
physical meaning and applications?
David Reutter Quantum graph isomorphisms March 20, 2018 4 / 23
![Page 15: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/15.jpg)
Quantum Functions
fSet ' (cfdC∗Alg)op ↪→ (fdC∗Alg)op
' fqSet{
finite setsfunctions
} commutative
f.d. C ∗-algebras∗-homomorphisms
op {
f.d. C ∗-algebras∗-homomorphisms
}op
set of functions fSet(A,B)
quantum set of maps fqSet(A,B)
How to ‘quantize’ morphisms?
Sketch-Definition:Quantum set of quantum functions ! internal hom [A,B] in C∗Algop
X ,Y finite sets. [X ,Y ] = universal C ∗-algebrawith generators {pxy}x∈X ,y∈Y and relations:
p∗xy = pxy = p2xy pxypxy ′ = δy ,y ′pxy∑y∈Y
pxy = 1
Variant: [1]Group of bijections Sn ! internal ‘automorphism group’ in C∗Algop
internal hom is infinite-dimensional operator algebras
compositional nature of quantum functions obscured
physical meaning and applications?
David Reutter Quantum graph isomorphisms March 20, 2018 4 / 23
![Page 16: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/16.jpg)
Quantum Functions
fSet ' (cfdC∗Alg)op ↪→ (fdC∗Alg)op ' fqSet
{finite setsfunctions
} commutative
f.d. C ∗-algebras∗-homomorphisms
op {
f.d. C ∗-algebras∗-homomorphisms
}op
set of functions fSet(A,B)
quantum set of maps fqSet(A,B)
How to ‘quantize’ morphisms?
Sketch-Definition:Quantum set of quantum functions ! internal hom [A,B] in C∗Algop
X ,Y finite sets. [X ,Y ] = universal C ∗-algebrawith generators {pxy}x∈X ,y∈Y and relations:
p∗xy = pxy = p2xy pxypxy ′ = δy ,y ′pxy∑y∈Y
pxy = 1
Variant: [1]Group of bijections Sn ! internal ‘automorphism group’ in C∗Algop
internal hom is infinite-dimensional operator algebras
compositional nature of quantum functions obscured
physical meaning and applications?
David Reutter Quantum graph isomorphisms March 20, 2018 4 / 23
![Page 17: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/17.jpg)
Quantum Functions
fSet ' (cfdC∗Alg)op ↪→ (fdC∗Alg)op ' fqSet
{finite setsfunctions
} commutative
f.d. C ∗-algebras∗-homomorphisms
op {
f.d. C ∗-algebras∗-homomorphisms
}op
set of functions fSet(A,B)
quantum set of maps fqSet(A,B)
How to ‘quantize’ morphisms?
Sketch-Definition:Quantum set of quantum functions ! internal hom [A,B] in C∗Algop
X ,Y finite sets. [X ,Y ] = universal C ∗-algebrawith generators {pxy}x∈X ,y∈Y and relations:
p∗xy = pxy = p2xy pxypxy ′ = δy ,y ′pxy∑y∈Y
pxy = 1
Variant: [1]Group of bijections Sn ! internal ‘automorphism group’ in C∗Algop
internal hom is infinite-dimensional operator algebras
compositional nature of quantum functions obscured
physical meaning and applications?
David Reutter Quantum graph isomorphisms March 20, 2018 4 / 23
![Page 18: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/18.jpg)
Quantum Functions
fSet ' (cfdC∗Alg)op ↪→ (fdC∗Alg)op ' fqSet
{finite setsfunctions
} commutative
f.d. C ∗-algebras∗-homomorphisms
op {
f.d. C ∗-algebras∗-homomorphisms
}op
set of functions fSet(A,B)
quantum
set of maps fqSet(A,B)
How to ‘quantize’ morphisms?
Sketch-Definition:Quantum set of quantum functions ! internal hom [A,B] in C∗Algop
X ,Y finite sets. [X ,Y ] = universal C ∗-algebrawith generators {pxy}x∈X ,y∈Y and relations:
p∗xy = pxy = p2xy pxypxy ′ = δy ,y ′pxy∑y∈Y
pxy = 1
Variant: [1]Group of bijections Sn ! internal ‘automorphism group’ in C∗Algop
internal hom is infinite-dimensional operator algebras
compositional nature of quantum functions obscured
physical meaning and applications?
David Reutter Quantum graph isomorphisms March 20, 2018 4 / 23
![Page 19: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/19.jpg)
Quantum Functions
fSet ' (cfdC∗Alg)op ↪→ (fdC∗Alg)op ' fqSet
{finite setsfunctions
} commutative
f.d. C ∗-algebras∗-homomorphisms
op {
f.d. C ∗-algebras∗-homomorphisms
}op
set of functions fSet(A,B) quantum set of maps fqSet(A,B)
How to ‘quantize’ morphisms?
Sketch-Definition:Quantum set of quantum functions ! internal hom [A,B] in C∗Algop
X ,Y finite sets. [X ,Y ] = universal C ∗-algebrawith generators {pxy}x∈X ,y∈Y and relations:
p∗xy = pxy = p2xy pxypxy ′ = δy ,y ′pxy∑y∈Y
pxy = 1
Variant: [1]Group of bijections Sn ! internal ‘automorphism group’ in C∗Algop
internal hom is infinite-dimensional operator algebras
compositional nature of quantum functions obscured
physical meaning and applications?
David Reutter Quantum graph isomorphisms March 20, 2018 4 / 23
![Page 20: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/20.jpg)
Quantum Functions
fSet ' (cfdC∗Alg)op ↪→ (fdC∗Alg)op ' fqSet
{finite setsfunctions
} commutative
f.d. C ∗-algebras∗-homomorphisms
op {
f.d. C ∗-algebras∗-homomorphisms
}op
set of functions fSet(A,B) quantum set of maps fqSet(A,B)
How to ‘quantize’ morphisms?
Sketch-Definition:Quantum set of quantum functions ! internal hom [A,B] in C∗Algop
X ,Y finite sets. [X ,Y ] = universal C ∗-algebrawith generators {pxy}x∈X ,y∈Y and relations:
p∗xy = pxy = p2xy pxypxy ′ = δy ,y ′pxy∑y∈Y
pxy = 1
Variant: [1]Group of bijections Sn ! internal ‘automorphism group’ in C∗Algop
internal hom is infinite-dimensional operator algebras
compositional nature of quantum functions obscured
physical meaning and applications?
David Reutter Quantum graph isomorphisms March 20, 2018 4 / 23
![Page 21: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/21.jpg)
Quantum Functions
fSet ' (cfdC∗Alg)op ↪→ (fdC∗Alg)op ' fqSet
{finite setsfunctions
} commutative
f.d. C ∗-algebras∗-homomorphisms
op {
f.d. C ∗-algebras∗-homomorphisms
}op
set of functions fSet(A,B) quantum set of maps fqSet(A,B)
How to ‘quantize’ morphisms?
Sketch-Definition:Quantum set of quantum functions ! internal hom [A,B] in C∗Algop
X ,Y finite sets. [X ,Y ] = universal C ∗-algebrawith generators {pxy}x∈X ,y∈Y and relations:
p∗xy = pxy = p2xy pxypxy ′ = δy ,y ′pxy∑y∈Y
pxy = 1
Variant: [1]Group of bijections Sn ! internal ‘automorphism group’ in C∗Algop
internal hom is infinite-dimensional operator algebras
compositional nature of quantum functions obscured
physical meaning and applications?
David Reutter Quantum graph isomorphisms March 20, 2018 4 / 23
![Page 22: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/22.jpg)
Quantum Functions
fSet ' (cfdC∗Alg)op ↪→ (fdC∗Alg)op ' fqSet
{finite setsfunctions
} commutative
f.d. C ∗-algebras∗-homomorphisms
op {
f.d. C ∗-algebras∗-homomorphisms
}op
set of functions fSet(A,B) quantum set of maps fqSet(A,B)
How to ‘quantize’ morphisms?
Sketch-Definition:Quantum set of quantum functions ! internal hom [A,B] in C∗Algop
X ,Y finite sets. [X ,Y ] = universal C ∗-algebrawith generators {pxy}x∈X ,y∈Y and relations:
p∗xy = pxy = p2xy pxypxy ′ = δy ,y ′pxy∑y∈Y
pxy = 1
Variant: [1]Group of bijections Sn ! internal ‘automorphism group’ in C∗Algop
internal hom is infinite-dimensional operator algebras
compositional nature of quantum functions obscured
physical meaning and applications?
David Reutter Quantum graph isomorphisms March 20, 2018 4 / 23
![Page 23: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/23.jpg)
Quantum Functions
fSet ' (cfdC∗Alg)op ↪→ (fdC∗Alg)op ' fqSet
{finite setsfunctions
} commutative
f.d. C ∗-algebras∗-homomorphisms
op {
f.d. C ∗-algebras∗-homomorphisms
}op
set of functions fSet(A,B) quantum set of maps fqSet(A,B)
How to ‘quantize’ morphisms?
Sketch-Definition:Quantum set of quantum functions ! internal hom [A,B] in C∗Algop
X ,Y finite sets. [X ,Y ] = universal C ∗-algebrawith generators {pxy}x∈X ,y∈Y and relations:
p∗xy = pxy = p2xy pxypxy ′ = δy ,y ′pxy∑y∈Y
pxy = 1
Variant: [1]Group of bijections Sn ! internal ‘automorphism group’ in C∗Algop
internal hom is infinite-dimensional operator algebras
compositional nature of quantum functions obscured
physical meaning and applications?
David Reutter Quantum graph isomorphisms March 20, 2018 4 / 23
[1] Wang — Quantum symmetry groups of finite spaces. 1998
![Page 24: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/24.jpg)
Quantum Functions
fSet ' (cfdC∗Alg)op ↪→ (fdC∗Alg)op ' fqSet
{finite setsfunctions
} commutative
f.d. C ∗-algebras∗-homomorphisms
op {
f.d. C ∗-algebras∗-homomorphisms
}op
set of functions fSet(A,B) quantum set of maps fqSet(A,B)
How to ‘quantize’ morphisms?
Sketch-Definition:Quantum set of quantum functions ! internal hom [A,B] in C∗Algop
X ,Y finite sets. [X ,Y ] = universal C ∗-algebrawith generators {pxy}x∈X ,y∈Y and relations:
p∗xy = pxy = p2xy pxypxy ′ = δy ,y ′pxy∑y∈Y
pxy = 1
Variant: [1]Group of bijections Sn ! internal ‘automorphism group’ in C∗Algop
internal hom is infinite-dimensional operator algebras
compositional nature of quantum functions obscured
physical meaning and applications?
David Reutter Quantum graph isomorphisms March 20, 2018 4 / 23
[1] Wang — Quantum symmetry groups of finite spaces. 1998
![Page 25: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/25.jpg)
Quantum Functions
fSet ' (cfdC∗Alg)op ↪→ (fdC∗Alg)op ' fqSet
{finite setsfunctions
} commutative
f.d. C ∗-algebras∗-homomorphisms
op {
f.d. C ∗-algebras∗-homomorphisms
}op
set of functions fSet(A,B) quantum set of maps fqSet(A,B)
How to ‘quantize’ morphisms?
Sketch-Definition:Quantum set of quantum functions ! internal hom [A,B] in C∗Algop
X ,Y finite sets. [X ,Y ] = universal C ∗-algebrawith generators {pxy}x∈X ,y∈Y and relations:
p∗xy = pxy = p2xy pxypxy ′ = δy ,y ′pxy∑y∈Y
pxy = 1
Variant: [1]Group of bijections Sn ! internal ‘automorphism group’ in C∗Algop
internal hom is infinite-dimensional operator algebras
compositional nature of quantum functions obscured
physical meaning and applications?
David Reutter Quantum graph isomorphisms March 20, 2018 4 / 23
[1] Wang — Quantum symmetry groups of finite spaces. 1998
![Page 26: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/26.jpg)
Quantum Functions
fSet ' (cfdC∗Alg)op ↪→ (fdC∗Alg)op ' fqSet
{finite setsfunctions
} commutative
f.d. C ∗-algebras∗-homomorphisms
op {
f.d. C ∗-algebras∗-homomorphisms
}op
set of functions fSet(A,B) quantum set of maps fqSet(A,B)
How to ‘quantize’ morphisms?
Sketch-Definition:Quantum set of quantum functions ! internal hom [A,B] in C∗Algop
X ,Y finite sets. [X ,Y ] = universal C ∗-algebrawith generators {pxy}x∈X ,y∈Y and relations:
p∗xy = pxy = p2xy pxypxy ′ = δy ,y ′pxy∑y∈Y
pxy = 1
Variant: [1]Group of bijections Sn ! internal ‘automorphism group’ in C∗Algop
internal hom is infinite-dimensional operator algebras
compositional nature of quantum functions obscured
physical meaning and applications?
David Reutter Quantum graph isomorphisms March 20, 2018 4 / 23
[1] Wang — Quantum symmetry groups of finite spaces. 1998
![Page 27: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/27.jpg)
Pseudo-telepathy and quantum graph isomorphisms
Pseudo-telepathy: Use entanglement to perform impossible tasks.
Definition (Graph isomorphism game [1])
Let G and H be graphs with vertex sets V (G ) and V (H).Alice and Bob play against a verifier.They cannot communicate once the game has started.Step 1: The verifier gives Alice and Bob vertices vA, vB ∈ V (G ) ∪ V (H).Step 2: They reply with vertices v ′A and v ′B in V (G ) ∪ V (H).Rules: Alice and Bob win if:
(i) vA ∈ V (G )⇔ v ′A ∈ V (H) and vB ∈ V (G )⇔ v ′B ∈ V (H)Let vGA = unique vertex of {vA, v ′A} in G and similarly vHA , v
GB , v
HB
(ii) vGA = vGB ⇔ vHA = vHB(iii) vGA ∼G vGB ⇔ vHA ∼H vHB
A perfect winning strategy is a graph isomorphisms.There are graphs that are quantum but not classically isomorphic!
Suppose Alice and Bob share a maximally entangled state on H⊗H.Quantum graph isomorphism = projectors {Pxy}x∈V (G),y∈V (H) on Hsuch that:
PxyPxy ′ = δy ,y ′Pxy
∑y∈V (H)
Pxy = idH
PxyPx ′y = δx ,x ′Pxy
∑x∈V (G)
Pxy = idH
If x , x ′ ∈ V (G ), y , y ′ ∈ V (H) with x ∼G x ′ XOR y ∼H y ′, then
PxyPx ′y ′ = 0
David Reutter Quantum graph isomorphisms March 20, 2018 5 / 23
![Page 28: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/28.jpg)
Pseudo-telepathy and quantum graph isomorphisms
Pseudo-telepathy: Use entanglement to perform impossible tasks.
Definition (Graph isomorphism game [1])
Let G and H be graphs with vertex sets V (G ) and V (H).Alice and Bob play against a verifier.They cannot communicate once the game has started.Step 1: The verifier gives Alice and Bob vertices vA, vB ∈ V (G ) ∪ V (H).Step 2: They reply with vertices v ′A and v ′B in V (G ) ∪ V (H).Rules: Alice and Bob win if:
(i) vA ∈ V (G )⇔ v ′A ∈ V (H) and vB ∈ V (G )⇔ v ′B ∈ V (H)Let vGA = unique vertex of {vA, v ′A} in G and similarly vHA , v
GB , v
HB
(ii) vGA = vGB ⇔ vHA = vHB(iii) vGA ∼G vGB ⇔ vHA ∼H vHB
A perfect winning strategy is a graph isomorphisms.There are graphs that are quantum but not classically isomorphic!
Suppose Alice and Bob share a maximally entangled state on H⊗H.Quantum graph isomorphism = projectors {Pxy}x∈V (G),y∈V (H) on Hsuch that:
PxyPxy ′ = δy ,y ′Pxy
∑y∈V (H)
Pxy = idH
PxyPx ′y = δx ,x ′Pxy
∑x∈V (G)
Pxy = idH
If x , x ′ ∈ V (G ), y , y ′ ∈ V (H) with x ∼G x ′ XOR y ∼H y ′, then
PxyPx ′y ′ = 0
David Reutter Quantum graph isomorphisms March 20, 2018 5 / 23
![Page 29: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/29.jpg)
Pseudo-telepathy and quantum graph isomorphisms
Pseudo-telepathy: Use entanglement to perform impossible tasks.
Definition (Graph isomorphism game [1])
Let G and H be graphs with vertex sets V (G ) and V (H).
Alice and Bob play against a verifier.They cannot communicate once the game has started.Step 1: The verifier gives Alice and Bob vertices vA, vB ∈ V (G ) ∪ V (H).Step 2: They reply with vertices v ′A and v ′B in V (G ) ∪ V (H).Rules: Alice and Bob win if:
(i) vA ∈ V (G )⇔ v ′A ∈ V (H) and vB ∈ V (G )⇔ v ′B ∈ V (H)Let vGA = unique vertex of {vA, v ′A} in G and similarly vHA , v
GB , v
HB
(ii) vGA = vGB ⇔ vHA = vHB(iii) vGA ∼G vGB ⇔ vHA ∼H vHB
A perfect winning strategy is a graph isomorphisms.There are graphs that are quantum but not classically isomorphic!
Suppose Alice and Bob share a maximally entangled state on H⊗H.Quantum graph isomorphism = projectors {Pxy}x∈V (G),y∈V (H) on Hsuch that:
PxyPxy ′ = δy ,y ′Pxy
∑y∈V (H)
Pxy = idH
PxyPx ′y = δx ,x ′Pxy
∑x∈V (G)
Pxy = idH
If x , x ′ ∈ V (G ), y , y ′ ∈ V (H) with x ∼G x ′ XOR y ∼H y ′, then
PxyPx ′y ′ = 0
David Reutter Quantum graph isomorphisms March 20, 2018 5 / 23
[1] Atserias, Mancinska, Roberson, Samal, Severini and Varvitsiotis — Quantum [...] graph isomorphisms. 2017
![Page 30: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/30.jpg)
Pseudo-telepathy and quantum graph isomorphisms
Pseudo-telepathy: Use entanglement to perform impossible tasks.
Definition (Graph isomorphism game [1])
Let G and H be graphs with vertex sets V (G ) and V (H).Alice and Bob play against a verifier.
They cannot communicate once the game has started.Step 1: The verifier gives Alice and Bob vertices vA, vB ∈ V (G ) ∪ V (H).Step 2: They reply with vertices v ′A and v ′B in V (G ) ∪ V (H).Rules: Alice and Bob win if:
(i) vA ∈ V (G )⇔ v ′A ∈ V (H) and vB ∈ V (G )⇔ v ′B ∈ V (H)Let vGA = unique vertex of {vA, v ′A} in G and similarly vHA , v
GB , v
HB
(ii) vGA = vGB ⇔ vHA = vHB(iii) vGA ∼G vGB ⇔ vHA ∼H vHB
A perfect winning strategy is a graph isomorphisms.There are graphs that are quantum but not classically isomorphic!
Suppose Alice and Bob share a maximally entangled state on H⊗H.Quantum graph isomorphism = projectors {Pxy}x∈V (G),y∈V (H) on Hsuch that:
PxyPxy ′ = δy ,y ′Pxy
∑y∈V (H)
Pxy = idH
PxyPx ′y = δx ,x ′Pxy
∑x∈V (G)
Pxy = idH
If x , x ′ ∈ V (G ), y , y ′ ∈ V (H) with x ∼G x ′ XOR y ∼H y ′, then
PxyPx ′y ′ = 0
David Reutter Quantum graph isomorphisms March 20, 2018 5 / 23
[1] Atserias, Mancinska, Roberson, Samal, Severini and Varvitsiotis — Quantum [...] graph isomorphisms. 2017
![Page 31: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/31.jpg)
Pseudo-telepathy and quantum graph isomorphisms
Pseudo-telepathy: Use entanglement to perform impossible tasks.
Definition (Graph isomorphism game [1])
Let G and H be graphs with vertex sets V (G ) and V (H).Alice and Bob play against a verifier.They cannot communicate once the game has started.
Step 1: The verifier gives Alice and Bob vertices vA, vB ∈ V (G ) ∪ V (H).Step 2: They reply with vertices v ′A and v ′B in V (G ) ∪ V (H).Rules: Alice and Bob win if:
(i) vA ∈ V (G )⇔ v ′A ∈ V (H) and vB ∈ V (G )⇔ v ′B ∈ V (H)Let vGA = unique vertex of {vA, v ′A} in G and similarly vHA , v
GB , v
HB
(ii) vGA = vGB ⇔ vHA = vHB(iii) vGA ∼G vGB ⇔ vHA ∼H vHB
A perfect winning strategy is a graph isomorphisms.There are graphs that are quantum but not classically isomorphic!
Suppose Alice and Bob share a maximally entangled state on H⊗H.Quantum graph isomorphism = projectors {Pxy}x∈V (G),y∈V (H) on Hsuch that:
PxyPxy ′ = δy ,y ′Pxy
∑y∈V (H)
Pxy = idH
PxyPx ′y = δx ,x ′Pxy
∑x∈V (G)
Pxy = idH
If x , x ′ ∈ V (G ), y , y ′ ∈ V (H) with x ∼G x ′ XOR y ∼H y ′, then
PxyPx ′y ′ = 0
David Reutter Quantum graph isomorphisms March 20, 2018 5 / 23
[1] Atserias, Mancinska, Roberson, Samal, Severini and Varvitsiotis — Quantum [...] graph isomorphisms. 2017
![Page 32: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/32.jpg)
Pseudo-telepathy and quantum graph isomorphisms
Pseudo-telepathy: Use entanglement to perform impossible tasks.
Definition (Graph isomorphism game [1])
Let G and H be graphs with vertex sets V (G ) and V (H).Alice and Bob play against a verifier.They cannot communicate once the game has started.Step 1: The verifier gives Alice and Bob vertices vA, vB ∈ V (G ) ∪ V (H).
Step 2: They reply with vertices v ′A and v ′B in V (G ) ∪ V (H).Rules: Alice and Bob win if:
(i) vA ∈ V (G )⇔ v ′A ∈ V (H) and vB ∈ V (G )⇔ v ′B ∈ V (H)Let vGA = unique vertex of {vA, v ′A} in G and similarly vHA , v
GB , v
HB
(ii) vGA = vGB ⇔ vHA = vHB(iii) vGA ∼G vGB ⇔ vHA ∼H vHB
A perfect winning strategy is a graph isomorphisms.There are graphs that are quantum but not classically isomorphic!
Suppose Alice and Bob share a maximally entangled state on H⊗H.Quantum graph isomorphism = projectors {Pxy}x∈V (G),y∈V (H) on Hsuch that:
PxyPxy ′ = δy ,y ′Pxy
∑y∈V (H)
Pxy = idH
PxyPx ′y = δx ,x ′Pxy
∑x∈V (G)
Pxy = idH
If x , x ′ ∈ V (G ), y , y ′ ∈ V (H) with x ∼G x ′ XOR y ∼H y ′, then
PxyPx ′y ′ = 0
David Reutter Quantum graph isomorphisms March 20, 2018 5 / 23
[1] Atserias, Mancinska, Roberson, Samal, Severini and Varvitsiotis — Quantum [...] graph isomorphisms. 2017
![Page 33: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/33.jpg)
Pseudo-telepathy and quantum graph isomorphisms
Pseudo-telepathy: Use entanglement to perform impossible tasks.
Definition (Graph isomorphism game [1])
Let G and H be graphs with vertex sets V (G ) and V (H).Alice and Bob play against a verifier.They cannot communicate once the game has started.Step 1: The verifier gives Alice and Bob vertices vA, vB ∈ V (G ) ∪ V (H).Step 2: They reply with vertices v ′A and v ′B in V (G ) ∪ V (H).
Rules: Alice and Bob win if:
(i) vA ∈ V (G )⇔ v ′A ∈ V (H) and vB ∈ V (G )⇔ v ′B ∈ V (H)Let vGA = unique vertex of {vA, v ′A} in G and similarly vHA , v
GB , v
HB
(ii) vGA = vGB ⇔ vHA = vHB(iii) vGA ∼G vGB ⇔ vHA ∼H vHB
A perfect winning strategy is a graph isomorphisms.There are graphs that are quantum but not classically isomorphic!
Suppose Alice and Bob share a maximally entangled state on H⊗H.Quantum graph isomorphism = projectors {Pxy}x∈V (G),y∈V (H) on Hsuch that:
PxyPxy ′ = δy ,y ′Pxy
∑y∈V (H)
Pxy = idH
PxyPx ′y = δx ,x ′Pxy
∑x∈V (G)
Pxy = idH
If x , x ′ ∈ V (G ), y , y ′ ∈ V (H) with x ∼G x ′ XOR y ∼H y ′, then
PxyPx ′y ′ = 0
David Reutter Quantum graph isomorphisms March 20, 2018 5 / 23
[1] Atserias, Mancinska, Roberson, Samal, Severini and Varvitsiotis — Quantum [...] graph isomorphisms. 2017
![Page 34: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/34.jpg)
Pseudo-telepathy and quantum graph isomorphisms
Pseudo-telepathy: Use entanglement to perform impossible tasks.
Definition (Graph isomorphism game [1])
Let G and H be graphs with vertex sets V (G ) and V (H).Alice and Bob play against a verifier.They cannot communicate once the game has started.Step 1: The verifier gives Alice and Bob vertices vA, vB ∈ V (G ) ∪ V (H).Step 2: They reply with vertices v ′A and v ′B in V (G ) ∪ V (H).Rules: Alice and Bob win if:
(i) vA ∈ V (G )⇔ v ′A ∈ V (H) and vB ∈ V (G )⇔ v ′B ∈ V (H)
Let vGA = unique vertex of {vA, v ′A} in G and similarly vHA , vGB , v
HB
(ii) vGA = vGB ⇔ vHA = vHB(iii) vGA ∼G vGB ⇔ vHA ∼H vHB
A perfect winning strategy is a graph isomorphisms.There are graphs that are quantum but not classically isomorphic!
Suppose Alice and Bob share a maximally entangled state on H⊗H.Quantum graph isomorphism = projectors {Pxy}x∈V (G),y∈V (H) on Hsuch that:
PxyPxy ′ = δy ,y ′Pxy
∑y∈V (H)
Pxy = idH
PxyPx ′y = δx ,x ′Pxy
∑x∈V (G)
Pxy = idH
If x , x ′ ∈ V (G ), y , y ′ ∈ V (H) with x ∼G x ′ XOR y ∼H y ′, then
PxyPx ′y ′ = 0
David Reutter Quantum graph isomorphisms March 20, 2018 5 / 23
[1] Atserias, Mancinska, Roberson, Samal, Severini and Varvitsiotis — Quantum [...] graph isomorphisms. 2017
![Page 35: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/35.jpg)
Pseudo-telepathy and quantum graph isomorphisms
Pseudo-telepathy: Use entanglement to perform impossible tasks.
Definition (Graph isomorphism game [1])
Let G and H be graphs with vertex sets V (G ) and V (H).Alice and Bob play against a verifier.They cannot communicate once the game has started.Step 1: The verifier gives Alice and Bob vertices vA, vB ∈ V (G ) ∪ V (H).Step 2: They reply with vertices v ′A and v ′B in V (G ) ∪ V (H).Rules: Alice and Bob win if:
(i) vA ∈ V (G )⇔ v ′A ∈ V (H) and vB ∈ V (G )⇔ v ′B ∈ V (H)Let vGA = unique vertex of {vA, v ′A} in G and similarly vHA , v
GB , v
HB
(ii) vGA = vGB ⇔ vHA = vHB
(iii) vGA ∼G vGB ⇔ vHA ∼H vHB
A perfect winning strategy is a graph isomorphisms.There are graphs that are quantum but not classically isomorphic!
Suppose Alice and Bob share a maximally entangled state on H⊗H.Quantum graph isomorphism = projectors {Pxy}x∈V (G),y∈V (H) on Hsuch that:
PxyPxy ′ = δy ,y ′Pxy
∑y∈V (H)
Pxy = idH
PxyPx ′y = δx ,x ′Pxy
∑x∈V (G)
Pxy = idH
If x , x ′ ∈ V (G ), y , y ′ ∈ V (H) with x ∼G x ′ XOR y ∼H y ′, then
PxyPx ′y ′ = 0
David Reutter Quantum graph isomorphisms March 20, 2018 5 / 23
[1] Atserias, Mancinska, Roberson, Samal, Severini and Varvitsiotis — Quantum [...] graph isomorphisms. 2017
![Page 36: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/36.jpg)
Pseudo-telepathy and quantum graph isomorphisms
Pseudo-telepathy: Use entanglement to perform impossible tasks.
Definition (Graph isomorphism game [1])
Let G and H be graphs with vertex sets V (G ) and V (H).Alice and Bob play against a verifier.They cannot communicate once the game has started.Step 1: The verifier gives Alice and Bob vertices vA, vB ∈ V (G ) ∪ V (H).Step 2: They reply with vertices v ′A and v ′B in V (G ) ∪ V (H).Rules: Alice and Bob win if:
(i) vA ∈ V (G )⇔ v ′A ∈ V (H) and vB ∈ V (G )⇔ v ′B ∈ V (H)Let vGA = unique vertex of {vA, v ′A} in G and similarly vHA , v
GB , v
HB
(ii) vGA = vGB ⇔ vHA = vHB(iii) vGA ∼G vGB ⇔ vHA ∼H vHB
A perfect winning strategy is a graph isomorphisms.There are graphs that are quantum but not classically isomorphic!
Suppose Alice and Bob share a maximally entangled state on H⊗H.Quantum graph isomorphism = projectors {Pxy}x∈V (G),y∈V (H) on Hsuch that:
PxyPxy ′ = δy ,y ′Pxy
∑y∈V (H)
Pxy = idH
PxyPx ′y = δx ,x ′Pxy
∑x∈V (G)
Pxy = idH
If x , x ′ ∈ V (G ), y , y ′ ∈ V (H) with x ∼G x ′ XOR y ∼H y ′, then
PxyPx ′y ′ = 0
David Reutter Quantum graph isomorphisms March 20, 2018 5 / 23
[1] Atserias, Mancinska, Roberson, Samal, Severini and Varvitsiotis — Quantum [...] graph isomorphisms. 2017
![Page 37: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/37.jpg)
Pseudo-telepathy and quantum graph isomorphisms
Pseudo-telepathy: Use entanglement to perform impossible tasks.
Definition (Graph isomorphism game [1])
Let G and H be graphs with vertex sets V (G ) and V (H).Alice and Bob play against a verifier.They cannot communicate once the game has started.Step 1: The verifier gives Alice and Bob vertices vA, vB ∈ V (G ) ∪ V (H).Step 2: They reply with vertices v ′A and v ′B in V (G ) ∪ V (H).Rules: Alice and Bob win if:
(i) vA ∈ V (G )⇔ v ′A ∈ V (H) and vB ∈ V (G )⇔ v ′B ∈ V (H)Let vGA = unique vertex of {vA, v ′A} in G and similarly vHA , v
GB , v
HB
(ii) vGA = vGB ⇔ vHA = vHB(iii) vGA ∼G vGB ⇔ vHA ∼H vHB
A perfect winning strategy is a graph isomorphisms.
There are graphs that are quantum but not classically isomorphic!
Suppose Alice and Bob share a maximally entangled state on H⊗H.Quantum graph isomorphism = projectors {Pxy}x∈V (G),y∈V (H) on Hsuch that:
PxyPxy ′ = δy ,y ′Pxy
∑y∈V (H)
Pxy = idH
PxyPx ′y = δx ,x ′Pxy
∑x∈V (G)
Pxy = idH
If x , x ′ ∈ V (G ), y , y ′ ∈ V (H) with x ∼G x ′ XOR y ∼H y ′, then
PxyPx ′y ′ = 0
David Reutter Quantum graph isomorphisms March 20, 2018 5 / 23
[1] Atserias, Mancinska, Roberson, Samal, Severini and Varvitsiotis — Quantum [...] graph isomorphisms. 2017
![Page 38: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/38.jpg)
Pseudo-telepathy and quantum graph isomorphisms
Pseudo-telepathy: Use entanglement to perform impossible tasks.
Definition (Quantum graph isomorphism game [1])
Let G and H be graphs with vertex sets V (G ) and V (H).Alice and Bob play against a verifier and share an entangled state.They cannot communicate once the game has started.Step 1: The verifier gives Alice and Bob vertices vA, vB ∈ V (G ) ∪ V (H).Step 2: They reply with vertices v ′A and v ′B in V (G ) ∪ V (H).Rules: Alice and Bob win if:
(i) vA ∈ V (G )⇔ v ′A ∈ V (H) and vB ∈ V (G )⇔ v ′B ∈ V (H)Let vGA = unique vertex of {vA, v ′A} in G and similarly vHA , v
GB , v
HB
(ii) vGA = vGB ⇔ vHA = vHB(iii) vGA ∼G vGB ⇔ vHA ∼H vHB
A perfect winning strategy is a quantum graph isomorphisms.
There are graphs that are quantum but not classically isomorphic!
Suppose Alice and Bob share a maximally entangled state on H⊗H.Quantum graph isomorphism = projectors {Pxy}x∈V (G),y∈V (H) on Hsuch that:
PxyPxy ′ = δy ,y ′Pxy
∑y∈V (H)
Pxy = idH
PxyPx ′y = δx ,x ′Pxy
∑x∈V (G)
Pxy = idH
If x , x ′ ∈ V (G ), y , y ′ ∈ V (H) with x ∼G x ′ XOR y ∼H y ′, then
PxyPx ′y ′ = 0
David Reutter Quantum graph isomorphisms March 20, 2018 5 / 23
[1] Atserias, Mancinska, Roberson, Samal, Severini and Varvitsiotis — Quantum [...] graph isomorphisms. 2017
![Page 39: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/39.jpg)
Pseudo-telepathy and quantum graph isomorphisms
Pseudo-telepathy: Use entanglement to perform impossible tasks.
Definition (Quantum graph isomorphism game [1])
Let G and H be graphs with vertex sets V (G ) and V (H).Alice and Bob play against a verifier and share an entangled state.They cannot communicate once the game has started.Step 1: The verifier gives Alice and Bob vertices vA, vB ∈ V (G ) ∪ V (H).Step 2: They reply with vertices v ′A and v ′B in V (G ) ∪ V (H).Rules: Alice and Bob win if:
(i) vA ∈ V (G )⇔ v ′A ∈ V (H) and vB ∈ V (G )⇔ v ′B ∈ V (H)Let vGA = unique vertex of {vA, v ′A} in G and similarly vHA , v
GB , v
HB
(ii) vGA = vGB ⇔ vHA = vHB(iii) vGA ∼G vGB ⇔ vHA ∼H vHB
A perfect winning strategy is a quantum graph isomorphisms.There are graphs that are quantum but not classically isomorphic!
Suppose Alice and Bob share a maximally entangled state on H⊗H.Quantum graph isomorphism = projectors {Pxy}x∈V (G),y∈V (H) on Hsuch that:
PxyPxy ′ = δy ,y ′Pxy
∑y∈V (H)
Pxy = idH
PxyPx ′y = δx ,x ′Pxy
∑x∈V (G)
Pxy = idH
If x , x ′ ∈ V (G ), y , y ′ ∈ V (H) with x ∼G x ′ XOR y ∼H y ′, then
PxyPx ′y ′ = 0
David Reutter Quantum graph isomorphisms March 20, 2018 5 / 23
[1] Atserias, Mancinska, Roberson, Samal, Severini and Varvitsiotis — Quantum [...] graph isomorphisms. 2017
![Page 40: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/40.jpg)
Pseudo-telepathy and quantum graph isomorphisms
Pseudo-telepathy: Use entanglement to perform impossible tasks.
Definition (Quantum graph isomorphism game [1])
Let G and H be graphs with vertex sets V (G ) and V (H).Alice and Bob play against a verifier and share an entangled state.They cannot communicate once the game has started.Step 1: The verifier gives Alice and Bob vertices vA, vB ∈ V (G ) ∪ V (H).Step 2: They reply with vertices v ′A and v ′B in V (G ) ∪ V (H).Rules: Alice and Bob win if:
(i) vA ∈ V (G )⇔ v ′A ∈ V (H) and vB ∈ V (G )⇔ v ′B ∈ V (H)Let vGA = unique vertex of {vA, v ′A} in G and similarly vHA , v
GB , v
HB
(ii) vGA = vGB ⇔ vHA = vHB(iii) vGA ∼G vGB ⇔ vHA ∼H vHB
A perfect winning strategy is a quantum graph isomorphisms.There are graphs that are quantum but not classically isomorphic!
Suppose Alice and Bob share a maximally entangled state on H⊗H.Quantum graph isomorphism = projectors {Pxy}x∈V (G),y∈V (H) on Hsuch that:
PxyPxy ′ = δy ,y ′Pxy
∑y∈V (H)
Pxy = idH
PxyPx ′y = δx ,x ′Pxy
∑x∈V (G)
Pxy = idH
If x , x ′ ∈ V (G ), y , y ′ ∈ V (H) with x ∼G x ′ XOR y ∼H y ′, then
PxyPx ′y ′ = 0
David Reutter Quantum graph isomorphisms March 20, 2018 5 / 23
[1] Atserias, Mancinska, Roberson, Samal, Severini and Varvitsiotis — Quantum [...] graph isomorphisms. 2017
![Page 41: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/41.jpg)
Setting the stage
The stage:Hilb — the category of finite-dimensional Hilbert spaces and linear maps.String diagrams: read from bottom to top.
Finite Gelfand duality: [1]
finite set X ! commutative finite-dimensional C ∗-algebra CX
The algebra structure copies and compares the elements of X :
:=∑x∈X
x† x†
x
:=∑x∈X
x
This makes CX into a commutative special †-Frobenius algebra in Hilb.
Philosophy: Do finite set theory with string diagrams in Hilb.
David Reutter Quantum graph isomorphisms March 20, 2018 6 / 23
![Page 42: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/42.jpg)
Setting the stage
The stage:Hilb — the category of finite-dimensional Hilbert spaces and linear maps.
String diagrams: read from bottom to top.
Finite Gelfand duality: [1]
finite set X ! commutative finite-dimensional C ∗-algebra CX
The algebra structure copies and compares the elements of X :
:=∑x∈X
x† x†
x
:=∑x∈X
x
This makes CX into a commutative special †-Frobenius algebra in Hilb.
Philosophy: Do finite set theory with string diagrams in Hilb.
David Reutter Quantum graph isomorphisms March 20, 2018 6 / 23
![Page 43: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/43.jpg)
Setting the stage
The stage:Hilb — the category of finite-dimensional Hilbert spaces and linear maps.String diagrams: read from bottom to top.
Finite Gelfand duality: [1]
finite set X ! commutative finite-dimensional C ∗-algebra CX
The algebra structure copies and compares the elements of X :
:=∑x∈X
x† x†
x
:=∑x∈X
x
This makes CX into a commutative special †-Frobenius algebra in Hilb.
Philosophy: Do finite set theory with string diagrams in Hilb.
David Reutter Quantum graph isomorphisms March 20, 2018 6 / 23
![Page 44: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/44.jpg)
Setting the stage
The stage:Hilb — the category of finite-dimensional Hilbert spaces and linear maps.String diagrams: read from bottom to top.
Finite Gelfand duality: [1]
finite set X ! commutative finite-dimensional C ∗-algebra CX
The algebra structure copies and compares the elements of X :
:=∑x∈X
x† x†
x
:=∑x∈X
x
This makes CX into a commutative special †-Frobenius algebra in Hilb.
Philosophy: Do finite set theory with string diagrams in Hilb.
David Reutter Quantum graph isomorphisms March 20, 2018 6 / 23
[1] Coecke, Pavlovic, Vicary — A new description of orthogonal bases. 2008
![Page 45: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/45.jpg)
Setting the stage
The stage:Hilb — the category of finite-dimensional Hilbert spaces and linear maps.String diagrams: read from bottom to top.
Finite Gelfand duality: [1]
finite set X ! commutative finite-dimensional C ∗-algebra CX
The algebra structure copies and compares the elements of X :
:=∑x∈X
x† x†
x
:=∑x∈X
x
This makes CX into a commutative special †-Frobenius algebra in Hilb.
Philosophy: Do finite set theory with string diagrams in Hilb.
David Reutter Quantum graph isomorphisms March 20, 2018 6 / 23
[1] Coecke, Pavlovic, Vicary — A new description of orthogonal bases. 2008
![Page 46: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/46.jpg)
Setting the stage
The stage:Hilb — the category of finite-dimensional Hilbert spaces and linear maps.String diagrams: read from bottom to top.
Finite Gelfand duality: [1]
finite set X ! commutative finite-dimensional C ∗-algebra CX
The algebra structure copies and compares the elements of X :
:=∑x∈X
x† x†
x
:=∑x∈X
x
This makes CX into a commutative special †-Frobenius algebra in Hilb.
Philosophy: Do finite set theory with string diagrams in Hilb.
David Reutter Quantum graph isomorphisms March 20, 2018 6 / 23
[1] Coecke, Pavlovic, Vicary — A new description of orthogonal bases. 2008
![Page 47: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/47.jpg)
Setting the stage
The stage:Hilb — the category of finite-dimensional Hilbert spaces and linear maps.String diagrams: read from bottom to top.
Finite Gelfand duality: [1]
finite set X ! commutative finite-dimensional C ∗-algebra CX
The algebra structure copies and compares the elements of X :
:=∑x∈X
x† x†
x
:=∑x∈X
x
This makes CX into a commutative special †-Frobenius algebra in Hilb.
Philosophy: Do finite set theory with string diagrams in Hilb.
David Reutter Quantum graph isomorphisms March 20, 2018 6 / 23
[1] Coecke, Pavlovic, Vicary — A new description of orthogonal bases. 2008
![Page 48: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/48.jpg)
Setting the stage
The stage:Hilb — the category of finite-dimensional Hilbert spaces and linear maps.String diagrams: read from bottom to top.
Finite Gelfand duality: [1]
finite set X ! commutative finite-dimensional C ∗-algebra CX
The algebra structure copies and compares the elements of X :
:=∑x∈X
x† x†
x
:=∑x∈X
x
This makes CX into a commutative special †-Frobenius algebra in Hilb.
Philosophy: Do finite set theory with string diagrams in Hilb.
David Reutter Quantum graph isomorphisms March 20, 2018 6 / 23
[1] Coecke, Pavlovic, Vicary — A new description of orthogonal bases. 2008
![Page 49: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/49.jpg)
Setting the stage
The stage:Hilb — the category of finite-dimensional Hilbert spaces and linear maps.String diagrams: read from bottom to top.
Finite Gelfand duality: [1]
finite set X ! commutative special †-Frobenius algebra CX in Hilb
The algebra structure copies and compares the elements of X :
:=∑x∈X
x† x†
x
:=∑x∈X
x
This makes CX into a commutative special †-Frobenius algebra in Hilb.
Philosophy: Do finite set theory with string diagrams in Hilb.
David Reutter Quantum graph isomorphisms March 20, 2018 6 / 23
[1] Coecke, Pavlovic, Vicary — A new description of orthogonal bases. 2008
![Page 50: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/50.jpg)
Setting the stage
The stage:Hilb — the category of finite-dimensional Hilbert spaces and linear maps.String diagrams: read from bottom to top.
Finite Gelfand duality: [1]
finite set X ! commutative special †-Frobenius algebra CX in Hilb
The algebra structure copies and compares the elements of X :
:=∑x∈X
x† x†
x
:=∑x∈X
x
This makes CX into a commutative special †-Frobenius algebra in Hilb.
Philosophy: Do finite set theory with string diagrams in Hilb.
David Reutter Quantum graph isomorphisms March 20, 2018 6 / 23
[1] Coecke, Pavlovic, Vicary — A new description of orthogonal bases. 2008
![Page 51: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/51.jpg)
Part 2Quantum functions
David Reutter Quantum graph isomorphisms March 20, 2018 7 / 23
![Page 52: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/52.jpg)
Quantizing functions
Function P between finite sets:
P =P
P
P = P† = P
δy ,y ′Pxy = PxyPxy ′∑y
Pxy = idH P†xy = Pxy
generalizes classical functions
Hilbert space wire enforces noncommutativity:
a
b=
b
a
a
b6=
b
a
turns elements of a set into elements of another set usingobservations on an underlying quantum system
Recipe:1) take concept or proof from finite set theory2) express it in terms of string diagrams in Hilb3) stick a wire through it
These look like the equations satisfied by a braiding.
David Reutter Quantum graph isomorphisms March 20, 2018 8 / 23
![Page 53: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/53.jpg)
Quantizing functions
Quantum function (H,P) between finite sets:
P =P
P
P = P† = P
δy ,y ′Pxy = PxyPxy ′∑y
Pxy = idH P†xy = Pxy
generalizes classical functions
Hilbert space wire enforces noncommutativity:
a
b=
b
a
a
b6=
b
a
turns elements of a set into elements of another set usingobservations on an underlying quantum system
Recipe:1) take concept or proof from finite set theory2) express it in terms of string diagrams in Hilb3) stick a wire through it
These look like the equations satisfied by a braiding.
David Reutter Quantum graph isomorphisms March 20, 2018 8 / 23
![Page 54: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/54.jpg)
Quantizing functions
Quantum function (H,P) between finite sets:
P =P
P
P = P† = P
δy ,y ′Pxy = PxyPxy ′∑y
Pxy = idH P†xy = Pxy
generalizes classical functions
Hilbert space wire enforces noncommutativity:
a
b=
b
a
a
b6=
b
a
turns elements of a set into elements of another set usingobservations on an underlying quantum system
Recipe:1) take concept or proof from finite set theory2) express it in terms of string diagrams in Hilb3) stick a wire through it
These look like the equations satisfied by a braiding.
David Reutter Quantum graph isomorphisms March 20, 2018 8 / 23
![Page 55: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/55.jpg)
Quantizing functions
Quantum function (H,P) between finite sets:
P =P
P
P = P† = P
δy ,y ′Pxy = PxyPxy ′∑y
Pxy = idH P†xy = Pxy
generalizes classical functions
Hilbert space wire enforces noncommutativity:
a
b=
b
a
a
b6=
b
a
turns elements of a set into elements of another set usingobservations on an underlying quantum system
Recipe:1) take concept or proof from finite set theory2) express it in terms of string diagrams in Hilb3) stick a wire through it
These look like the equations satisfied by a braiding.
David Reutter Quantum graph isomorphisms March 20, 2018 8 / 23
![Page 56: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/56.jpg)
Quantizing functions
Quantum function (H,P) between finite sets:
P =P
P
P = P† = P
δy ,y ′Pxy = PxyPxy ′∑y
Pxy = idH P†xy = Pxy
generalizes classical functions
Hilbert space wire enforces noncommutativity:
a
b=
b
a
a
b6=
b
a
turns elements of a set into elements of another set usingobservations on an underlying quantum system
Recipe:1) take concept or proof from finite set theory2) express it in terms of string diagrams in Hilb3) stick a wire through it
These look like the equations satisfied by a braiding.
David Reutter Quantum graph isomorphisms March 20, 2018 8 / 23
![Page 57: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/57.jpg)
Quantizing functions
Quantum function (H,P) between finite sets:
P =P
P
P = P† = P
δy ,y ′Pxy = PxyPxy ′∑y
Pxy = idH P†xy = Pxy
generalizes classical functions
Hilbert space wire enforces noncommutativity:
a
b=
b
a
a
b6=
b
a
turns elements of a set into elements of another set
usingobservations on an underlying quantum system
Recipe:1) take concept or proof from finite set theory2) express it in terms of string diagrams in Hilb3) stick a wire through it
These look like the equations satisfied by a braiding.
David Reutter Quantum graph isomorphisms March 20, 2018 8 / 23
![Page 58: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/58.jpg)
Quantizing functions
Quantum function (H,P) between finite sets:
P =P
P
P = P† = P
δy ,y ′Pxy = PxyPxy ′∑y
Pxy = idH P†xy = Pxy
generalizes classical functions
Hilbert space wire enforces noncommutativity:
a
b=
b
a
a
b6=
b
a
turns elements of a set into elements of another set usingobservations on an underlying quantum system
Recipe:1) take concept or proof from finite set theory2) express it in terms of string diagrams in Hilb3) stick a wire through it
These look like the equations satisfied by a braiding.
David Reutter Quantum graph isomorphisms March 20, 2018 8 / 23
![Page 59: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/59.jpg)
Quantizing functions
Quantum function (H,P) between finite sets:
P =P
P
P = P† = P
δy ,y ′Pxy = PxyPxy ′∑y
Pxy = idH P†xy = Pxy
generalizes classical functions
Hilbert space wire enforces noncommutativity:
a
b=
b
a
a
b6=
b
a
turns elements of a set into elements of another set usingobservations on an underlying quantum system
Recipe:1) take concept or proof from finite set theory2) express it in terms of string diagrams in Hilb3) stick a wire through it
These look like the equations satisfied by a braiding.
David Reutter Quantum graph isomorphisms March 20, 2018 8 / 23
![Page 60: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/60.jpg)
Quantizing functions
Quantum function (H,P) between finite sets:
= = =
δy ,y ′Pxy = PxyPxy ′∑y
Pxy = idH P†xy = Pxy
generalizes classical functions
Hilbert space wire enforces noncommutativity:
a
b=
b
a
a
b6=
b
a
turns elements of a set into elements of another set usingobservations on an underlying quantum system
Recipe:1) take concept or proof from finite set theory2) express it in terms of string diagrams in Hilb3) stick a wire through it
These look like the equations satisfied by a braiding.David Reutter Quantum graph isomorphisms March 20, 2018 8 / 23
![Page 61: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/61.jpg)
Quantization ⇒ Categorification
This new definition has room for higher structure.
An intertwiner of quantum functions (H,P) =⇒ (H′,Q) is:
a linear map f : H −→ H′ such thatf
P=
f
Q
no interesting intertwiners between classical functions
keep track of change on underlying system
Set(A,B) : Set of functions between finite sets A and B
Connection to previous work in noncommutative topology:QSet(A,B) is the category of f.d. representations of the internal hom [A,B]
David Reutter Quantum graph isomorphisms March 20, 2018 9 / 23
![Page 62: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/62.jpg)
Quantization ⇒ Categorification
This new definition has room for higher structure.An intertwiner of quantum functions (H,P) =⇒ (H′,Q) is:
a linear map f : H −→ H′ such thatf
P=
f
Q
no interesting intertwiners between classical functions
keep track of change on underlying system
Set(A,B) : Set of functions between finite sets A and B
Connection to previous work in noncommutative topology:QSet(A,B) is the category of f.d. representations of the internal hom [A,B]
David Reutter Quantum graph isomorphisms March 20, 2018 9 / 23
![Page 63: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/63.jpg)
Quantization ⇒ Categorification
This new definition has room for higher structure.An intertwiner of quantum functions (H,P) =⇒ (H′,Q) is:
a linear map f : H −→ H′ such thatf
P=
f
Q
no interesting intertwiners between classical functions
keep track of change on underlying system
Set(A,B) : Set of functions between finite sets A and B
Connection to previous work in noncommutative topology:QSet(A,B) is the category of f.d. representations of the internal hom [A,B]
David Reutter Quantum graph isomorphisms March 20, 2018 9 / 23
![Page 64: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/64.jpg)
Quantization ⇒ Categorification
This new definition has room for higher structure.An intertwiner of quantum functions (H,P) =⇒ (H′,Q) is:
a linear map f : H −→ H′ such thatf
P=
f
Q
no interesting intertwiners between classical functions
keep track of change on underlying system
Set(A,B) : Set of functions between finite sets A and B
Connection to previous work in noncommutative topology:QSet(A,B) is the category of f.d. representations of the internal hom [A,B]
David Reutter Quantum graph isomorphisms March 20, 2018 9 / 23
![Page 65: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/65.jpg)
Quantization ⇒ Categorification
This new definition has room for higher structure.An intertwiner of quantum functions (H,P) =⇒ (H′,Q) is:
a linear map f : H −→ H′ such thatf
P=
f
Q
no interesting intertwiners between classical functions
keep track of change on underlying system
Set(A,B) : Set of functions between finite sets A and B
Connection to previous work in noncommutative topology:QSet(A,B) is the category of f.d. representations of the internal hom [A,B]
David Reutter Quantum graph isomorphisms March 20, 2018 9 / 23
![Page 66: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/66.jpg)
Quantization ⇒ Categorification
This new definition has room for higher structure.An intertwiner of quantum functions (H,P) =⇒ (H′,Q) is:
a linear map f : H −→ H′ such thatf
P=
f
Q
no interesting intertwiners between classical functions
keep track of change on underlying system
QSet(A,B) : Category of quantum functions between finite sets A and B
Connection to previous work in noncommutative topology:QSet(A,B) is the category of f.d. representations of the internal hom [A,B]
David Reutter Quantum graph isomorphisms March 20, 2018 9 / 23
![Page 67: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/67.jpg)
Quantization ⇒ Categorification
This new definition has room for higher structure.An intertwiner of quantum functions (H,P) =⇒ (H′,Q) is:
a linear map f : H −→ H′ such thatf
P=
f
Q
no interesting intertwiners between classical functions
keep track of change on underlying system
QSet(A,B) : Category of quantum functions between finite sets A and B
Connection to previous work in noncommutative topology:QSet(A,B) is the category of f.d. representations of the internal hom [A,B]
David Reutter Quantum graph isomorphisms March 20, 2018 9 / 23
![Page 68: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/68.jpg)
The 2-category QSet
Definition
The 2-category QSet is built from the following structures:
objects are finite sets A,B, ...;
1-morphisms A −→ B are quantum functions (H,P) : A −→ B;
2-morphisms (H,P) −→ (H ′,P ′) are intertwiners
The composition of two quantum functions (H,P) : A −→ B and(H ′,Q) : B −→ C is a quantum function (H ⊗H ′,Q ◦P) defined as follows:
Q ◦ P
H ⊗ H′
:=
H H′
P
Q
2-morphisms compose by tensor product and composition of linear maps.
Can be extended to also include quantum sets as objects.
David Reutter Quantum graph isomorphisms March 20, 2018 10 / 23
![Page 69: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/69.jpg)
The 2-category QSet
Definition
The 2-category QSet is built from the following structures:
objects are finite sets A,B, ...;
1-morphisms A −→ B are quantum functions (H,P) : A −→ B;
2-morphisms (H,P) −→ (H ′,P ′) are intertwiners
The composition of two quantum functions (H,P) : A −→ B and(H ′,Q) : B −→ C is a quantum function (H ⊗H ′,Q ◦P) defined as follows:
Q ◦ P
H ⊗ H′
:=
H H′
P
Q
2-morphisms compose by tensor product and composition of linear maps.
Can be extended to also include quantum sets as objects.
David Reutter Quantum graph isomorphisms March 20, 2018 10 / 23
![Page 70: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/70.jpg)
The 2-category QSet
Definition
The 2-category QSet is built from the following structures:
objects are finite sets A,B, ...;
1-morphisms A −→ B are quantum functions (H,P) : A −→ B;
2-morphisms (H,P) −→ (H ′,P ′) are intertwiners
The composition of two quantum functions (H,P) : A −→ B and(H ′,Q) : B −→ C is a quantum function (H ⊗H ′,Q ◦P) defined as follows:
Q ◦ P
H ⊗ H′
:=
H H′
P
Q
2-morphisms compose by tensor product and composition of linear maps.
Can be extended to also include quantum sets as objects.David Reutter Quantum graph isomorphisms March 20, 2018 10 / 23
![Page 71: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/71.jpg)
Quantum bijections
Function P between finite sets:
P =P
P
P = P† = P
P=
PP
P=
δy ,y ′Pxy = PxyPxy ′∑y
Pxy = idH P†xy = Pxy
δx ,x ′Pxy = PxyPx ′y
∑x
Px ,y = idH
Theorem. Equivalences in QSet are ordinary bijections.
Theorem. Quantum bijections are dagger dualizable quantum functions.
David Reutter Quantum graph isomorphisms March 20, 2018 11 / 23
![Page 72: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/72.jpg)
Quantum bijections
Bijection P between finite sets:
P =P
P
P = P† = P
P=
PP
P=
δy ,y ′Pxy = PxyPxy ′∑y
Pxy = idH P†xy = Pxy
δx ,x ′Pxy = PxyPx ′y
∑x
Px ,y = idH
Theorem. Equivalences in QSet are ordinary bijections.
Theorem. Quantum bijections are dagger dualizable quantum functions.
David Reutter Quantum graph isomorphisms March 20, 2018 11 / 23
![Page 73: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/73.jpg)
Quantum bijections
Quantum bijection (H,P) between finite sets:
P =P
P
P = P† = P
P=
PP
P=
δy ,y ′Pxy = PxyPxy ′∑y
Pxy = idH P†xy = Pxy
δx ,x ′Pxy = PxyPx ′y
∑x
Px ,y = idH
Theorem. Equivalences in QSet are ordinary bijections.
Theorem. Quantum bijections are dagger dualizable quantum functions.
David Reutter Quantum graph isomorphisms March 20, 2018 11 / 23
![Page 74: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/74.jpg)
Quantum bijections
Quantum bijection (H,P) between finite sets:
= = =
= =
δy ,y ′Pxy = PxyPxy ′∑y
Pxy = idH P†xy = Pxy
δx ,x ′Pxy = PxyPx ′y
∑x
Px ,y = idH
Theorem. Equivalences in QSet are ordinary bijections.
Theorem. Quantum bijections are dagger dualizable quantum functions.
David Reutter Quantum graph isomorphisms March 20, 2018 11 / 23
![Page 75: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/75.jpg)
Quantum bijections
Quantum bijection (H,P) between finite sets:
P =P
P
P = P† = P
P=
PP
P=
δy ,y ′Pxy = PxyPxy ′∑y
Pxy = idH P†xy = Pxy
δx ,x ′Pxy = PxyPx ′y
∑x
Px ,y = idH
Theorem. Equivalences in QSet are ordinary bijections.
Theorem. Quantum bijections are dagger dualizable quantum functions.
David Reutter Quantum graph isomorphisms March 20, 2018 11 / 23
![Page 76: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/76.jpg)
Quantum bijections
Quantum bijection (H,P) between finite sets:
P =P
P
P = P† = P
P=
PP
P=
δy ,y ′Pxy = PxyPxy ′∑y
Pxy = idH P†xy = Pxy
δx ,x ′Pxy = PxyPx ′y
∑x
Px ,y = idH
Theorem. Equivalences in QSet are ordinary bijections.
Theorem. Quantum bijections are dagger dualizable quantum functions.
David Reutter Quantum graph isomorphisms March 20, 2018 11 / 23
![Page 77: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/77.jpg)
Quantum bijections
Quantum bijection (H,P) between finite sets:
P =P
P
P = P† = P
P=
PP
P=
δy ,y ′Pxy = PxyPxy ′∑y
Pxy = idH P†xy = Pxy
δx ,x ′Pxy = PxyPx ′y
∑x
Px ,y = idH
Theorem. Equivalences in QSet are ordinary bijections.
Theorem. Quantum bijections are dagger dualizable quantum functions.David Reutter Quantum graph isomorphisms March 20, 2018 11 / 23
![Page 78: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/78.jpg)
Quantum graph isomorphisms
Let G and H be finite graphs with adjacency matrices G and H.
A graph isomorphism is a bijection P such that:
P
G= P
H
These are exactly the quantum graph isomorphisms from pseudo-telepathy.
Definition
The 2-category QGraph is built from the following structures:
objects are finite graphs G ,H, ...;
1-morphisms G −→ H are quantum graph isomorphisms;
2-morphisms are intertwiners
Quantum graph isomorphisms are dualizable 1-morphisms.
David Reutter Quantum graph isomorphisms March 20, 2018 12 / 23
![Page 79: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/79.jpg)
Quantum graph isomorphisms
Let G and H be finite graphs with adjacency matrices G and H.A graph isomorphism is a bijection P such that:
P
G= P
H
These are exactly the quantum graph isomorphisms from pseudo-telepathy.
Definition
The 2-category QGraph is built from the following structures:
objects are finite graphs G ,H, ...;
1-morphisms G −→ H are quantum graph isomorphisms;
2-morphisms are intertwiners
Quantum graph isomorphisms are dualizable 1-morphisms.
David Reutter Quantum graph isomorphisms March 20, 2018 12 / 23
![Page 80: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/80.jpg)
Quantum graph isomorphisms
Let G and H be finite graphs with adjacency matrices G and H.A quantum graph isomorphism is a quantum bijection P such that:
P
G= P
H
These are exactly the quantum graph isomorphisms from pseudo-telepathy.
Definition
The 2-category QGraph is built from the following structures:
objects are finite graphs G ,H, ...;
1-morphisms G −→ H are quantum graph isomorphisms;
2-morphisms are intertwiners
Quantum graph isomorphisms are dualizable 1-morphisms.
David Reutter Quantum graph isomorphisms March 20, 2018 12 / 23
![Page 81: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/81.jpg)
Quantum graph isomorphisms
Let G and H be finite graphs with adjacency matrices G and H.A quantum graph isomorphism is a quantum bijection P such that:
P
G= P
H
These are exactly the quantum graph isomorphisms from pseudo-telepathy.
Definition
The 2-category QGraph is built from the following structures:
objects are finite graphs G ,H, ...;
1-morphisms G −→ H are quantum graph isomorphisms;
2-morphisms are intertwiners
Quantum graph isomorphisms are dualizable 1-morphisms.
David Reutter Quantum graph isomorphisms March 20, 2018 12 / 23
![Page 82: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/82.jpg)
Quantum graph isomorphisms
Let G and H be finite graphs with adjacency matrices G and H.A quantum graph isomorphism is a quantum bijection P such that:
P
G= P
H
These are exactly the quantum graph isomorphisms from pseudo-telepathy.
Definition
The 2-category QGraph is built from the following structures:
objects are finite graphs G ,H, ...;
1-morphisms G −→ H are quantum graph isomorphisms;
2-morphisms are intertwiners
Quantum graph isomorphisms are dualizable 1-morphisms.
David Reutter Quantum graph isomorphisms March 20, 2018 12 / 23
![Page 83: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/83.jpg)
Quantum graph isomorphisms
Let G and H be finite graphs with adjacency matrices G and H.A quantum graph isomorphism is a quantum bijection P such that:
P
G= P
H
These are exactly the quantum graph isomorphisms from pseudo-telepathy.
Definition
The 2-category QGraph is built from the following structures:
objects are finite graphs G ,H, ...;
1-morphisms G −→ H are quantum graph isomorphisms;
2-morphisms are intertwiners
Quantum graph isomorphisms are dualizable 1-morphisms.
David Reutter Quantum graph isomorphisms March 20, 2018 12 / 23
![Page 84: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/84.jpg)
QGraph
QGraph is a semisimple dagger 2-category with dualizable 1-morphisms.
Can take direct sum of quantum functions.
Every quantum function is a direct sum of simple quantum functionsP for which the intertwiners are trivial; QGraph(P,P) ∼= C.
QAut(G ) := QGraph(G ,G ) — the quantum automorphism category of agraph G — is a fusion category.
Every classical isomorphism is simple.
Direct sums of isomorphisms form the classical subcategory Aut(G ).
Aut(G ) = HilbAut(G), the category of Aut(G )-graded Hilbert spaces.
If QAut(G ) = Aut(G ), then G has no quantum symmetries.
David Reutter Quantum graph isomorphisms March 20, 2018 13 / 23
![Page 85: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/85.jpg)
QGraph
QGraph is a semisimple dagger 2-category with dualizable 1-morphisms.
Can take direct sum of quantum functions.
Every quantum function is a direct sum of simple quantum functionsP for which the intertwiners are trivial; QGraph(P,P) ∼= C.
QAut(G ) := QGraph(G ,G ) — the quantum automorphism category of agraph G — is a fusion category.
Every classical isomorphism is simple.
Direct sums of isomorphisms form the classical subcategory Aut(G ).
Aut(G ) = HilbAut(G), the category of Aut(G )-graded Hilbert spaces.
If QAut(G ) = Aut(G ), then G has no quantum symmetries.
David Reutter Quantum graph isomorphisms March 20, 2018 13 / 23
![Page 86: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/86.jpg)
QGraph
QGraph is a semisimple dagger 2-category with dualizable 1-morphisms.
Can take direct sum of quantum functions.
Every quantum function is a direct sum of simple quantum functionsP for which the intertwiners are trivial; QGraph(P,P) ∼= C.
QAut(G ) := QGraph(G ,G ) — the quantum automorphism category of agraph G — is a fusion category.
Every classical isomorphism is simple.
Direct sums of isomorphisms form the classical subcategory Aut(G ).
Aut(G ) = HilbAut(G), the category of Aut(G )-graded Hilbert spaces.
If QAut(G ) = Aut(G ), then G has no quantum symmetries.
David Reutter Quantum graph isomorphisms March 20, 2018 13 / 23
![Page 87: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/87.jpg)
QGraph
QGraph is a semisimple dagger 2-category with dualizable 1-morphisms.
Can take direct sum of quantum functions.
Every quantum function is a direct sum of simple quantum functionsP for which the intertwiners are trivial; QGraph(P,P) ∼= C.
QAut(G ) := QGraph(G ,G ) — the quantum automorphism category of agraph G — is a fusion1 category.
Every classical isomorphism is simple.
Direct sums of isomorphisms form the classical subcategory Aut(G ).
Aut(G ) = HilbAut(G), the category of Aut(G )-graded Hilbert spaces.
If QAut(G ) = Aut(G ), then G has no quantum symmetries.
1With possibly infinitely many simple objectsDavid Reutter Quantum graph isomorphisms March 20, 2018 13 / 23
![Page 88: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/88.jpg)
QGraph
QGraph is a semisimple dagger 2-category with dualizable 1-morphisms.
Can take direct sum of quantum functions.
Every quantum function is a direct sum of simple quantum functionsP for which the intertwiners are trivial; QGraph(P,P) ∼= C.
QAut(G ) := QGraph(G ,G ) — the quantum automorphism category of agraph G — is a fusion1 category.
Every classical isomorphism is simple.
Direct sums of isomorphisms form the classical subcategory Aut(G ).
Aut(G ) = HilbAut(G), the category of Aut(G )-graded Hilbert spaces.
If QAut(G ) = Aut(G ), then G has no quantum symmetries.
1With possibly infinitely many simple objectsDavid Reutter Quantum graph isomorphisms March 20, 2018 13 / 23
![Page 89: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/89.jpg)
QGraph
QGraph is a semisimple dagger 2-category with dualizable 1-morphisms.
Can take direct sum of quantum functions.
Every quantum function is a direct sum of simple quantum functionsP for which the intertwiners are trivial; QGraph(P,P) ∼= C.
QAut(G ) := QGraph(G ,G ) — the quantum automorphism category of agraph G — is a fusion1 category.
Every classical isomorphism is simple.
Direct sums of isomorphisms form the classical subcategory Aut(G ).
Aut(G ) = HilbAut(G), the category of Aut(G )-graded Hilbert spaces.
If QAut(G ) = Aut(G ), then G has no quantum symmetries.
1With possibly infinitely many simple objectsDavid Reutter Quantum graph isomorphisms March 20, 2018 13 / 23
![Page 90: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/90.jpg)
QGraph
QGraph is a semisimple dagger 2-category with dualizable 1-morphisms.
Can take direct sum of quantum functions.
Every quantum function is a direct sum of simple quantum functionsP for which the intertwiners are trivial; QGraph(P,P) ∼= C.
QAut(G ) := QGraph(G ,G ) — the quantum automorphism category of agraph G — is a fusion1 category.
Every classical isomorphism is simple.
Direct sums of isomorphisms form the classical subcategory Aut(G ).
Aut(G ) = HilbAut(G), the category of Aut(G )-graded Hilbert spaces.
If QAut(G ) = Aut(G ), then G has no quantum symmetries.
1With possibly infinitely many simple objectsDavid Reutter Quantum graph isomorphisms March 20, 2018 13 / 23
![Page 91: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/91.jpg)
QGraph
QGraph is a semisimple dagger 2-category with dualizable 1-morphisms.
Can take direct sum of quantum functions.
Every quantum function is a direct sum of simple quantum functionsP for which the intertwiners are trivial; QGraph(P,P) ∼= C.
QAut(G ) := QGraph(G ,G ) — the quantum automorphism category of agraph G — is a fusion1 category.
Every classical isomorphism is simple.
Direct sums of isomorphisms form the classical subcategory Aut(G ).
Aut(G ) = HilbAut(G), the category of Aut(G )-graded Hilbert spaces.
If QAut(G ) = Aut(G ), then G has no quantum symmetries.
1With possibly infinitely many simple objectsDavid Reutter Quantum graph isomorphisms March 20, 2018 13 / 23
![Page 92: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/92.jpg)
At the crossroads
Quantum automorphism groups of graphs have been studied before in thesetting of compact quantum groups [1] Hopf C ∗-algebra A(G )
Our QAut(G ) is the category of f.d. representations of A(G ).
We are now at the intersection of:
higher algebra: QAut(G ) is a fusion category.
compact quantum group theory: QAut(G ) = Rep(A(G ))
pseudo-telepathy: quantum but not classically isomorphic graphs
Can we understand quantum isomorphisms in terms of the quantumautomorphism categories QAut(G )?
David Reutter Quantum graph isomorphisms March 20, 2018 14 / 23
[1] Banica, Bichon and others — Quantum automorphism groups of graphs. 1999–
![Page 93: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/93.jpg)
At the crossroads
Quantum automorphism groups of graphs have been studied before in thesetting of compact quantum groups [1] Hopf C ∗-algebra A(G )Our QAut(G ) is the category of f.d. representations of A(G ).
We are now at the intersection of:
higher algebra: QAut(G ) is a fusion category.
compact quantum group theory: QAut(G ) = Rep(A(G ))
pseudo-telepathy: quantum but not classically isomorphic graphs
Can we understand quantum isomorphisms in terms of the quantumautomorphism categories QAut(G )?
David Reutter Quantum graph isomorphisms March 20, 2018 14 / 23
[1] Banica, Bichon and others — Quantum automorphism groups of graphs. 1999–
![Page 94: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/94.jpg)
At the crossroads
Quantum automorphism groups of graphs have been studied before in thesetting of compact quantum groups [1] Hopf C ∗-algebra A(G )Our QAut(G ) is the category of f.d. representations of A(G ).
We are now at the intersection of:
higher algebra: QAut(G ) is a fusion category.
compact quantum group theory: QAut(G ) = Rep(A(G ))
pseudo-telepathy: quantum but not classically isomorphic graphs
Can we understand quantum isomorphisms in terms of the quantumautomorphism categories QAut(G )?
David Reutter Quantum graph isomorphisms March 20, 2018 14 / 23
[1] Banica, Bichon and others — Quantum automorphism groups of graphs. 1999–
![Page 95: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/95.jpg)
At the crossroads
Quantum automorphism groups of graphs have been studied before in thesetting of compact quantum groups [1] Hopf C ∗-algebra A(G )Our QAut(G ) is the category of f.d. representations of A(G ).
We are now at the intersection of:
higher algebra: QAut(G ) is a fusion category.
compact quantum group theory: QAut(G ) = Rep(A(G ))
pseudo-telepathy: quantum but not classically isomorphic graphs
Can we understand quantum isomorphisms in terms of the quantumautomorphism categories QAut(G )?
David Reutter Quantum graph isomorphisms March 20, 2018 14 / 23
[1] Banica, Bichon and others — Quantum automorphism groups of graphs. 1999–
![Page 96: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/96.jpg)
At the crossroads
Quantum automorphism groups of graphs have been studied before in thesetting of compact quantum groups [1] Hopf C ∗-algebra A(G )Our QAut(G ) is the category of f.d. representations of A(G ).
We are now at the intersection of:
higher algebra: QAut(G ) is a fusion category.
compact quantum group theory: QAut(G ) = Rep(A(G ))
pseudo-telepathy: quantum but not classically isomorphic graphs
Can we understand quantum isomorphisms in terms of the quantumautomorphism categories QAut(G )?
David Reutter Quantum graph isomorphisms March 20, 2018 14 / 23
[1] Banica, Bichon and others — Quantum automorphism groups of graphs. 1999–
![Page 97: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/97.jpg)
At the crossroads
Quantum automorphism groups of graphs have been studied before in thesetting of compact quantum groups [1] Hopf C ∗-algebra A(G )Our QAut(G ) is the category of f.d. representations of A(G ).
We are now at the intersection of:
higher algebra: QAut(G ) is a fusion category.
compact quantum group theory: QAut(G ) = Rep(A(G ))
pseudo-telepathy: quantum but not classically isomorphic graphs
Can we understand quantum isomorphisms in terms of the quantumautomorphism categories QAut(G )?
David Reutter Quantum graph isomorphisms March 20, 2018 14 / 23
[1] Banica, Bichon and others — Quantum automorphism groups of graphs. 1999–
![Page 98: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/98.jpg)
Part 3The Morita theory of
quantum graph isomorphisms
David Reutter Quantum graph isomorphisms March 20, 2018 15 / 23
![Page 99: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/99.jpg)
Morita theory on a slide
Morita theory: The study of algebras up to Morita equivalence.
Definition
Algebras A,B are Morita equivalent ⇔the module categories Mod(A) and Mod(B) are equivalent
isomorphic algebras are Morita equivalent
Morita equivalent commutative algebras are isomorphic
Morita equivalent but non-isomorphic algebras: e.g. Matn(C) and C
Morita equivalence can be defined in arbitrary monoidal categories.
Morita classes of Frobenius algebras in fusion categories are well studied.
David Reutter Quantum graph isomorphisms March 20, 2018 16 / 23
![Page 100: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/100.jpg)
Morita theory on a slide
Morita theory: The study of algebras up to Morita equivalence.
Definition
Algebras A,B are Morita equivalent ⇔the module categories Mod(A) and Mod(B) are equivalent
isomorphic algebras are Morita equivalent
Morita equivalent commutative algebras are isomorphic
Morita equivalent but non-isomorphic algebras: e.g. Matn(C) and C
Morita equivalence can be defined in arbitrary monoidal categories.
Morita classes of Frobenius algebras in fusion categories are well studied.
David Reutter Quantum graph isomorphisms March 20, 2018 16 / 23
![Page 101: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/101.jpg)
Morita theory on a slide
Morita theory: The study of algebras up to Morita equivalence.
Definition
Algebras A,B are Morita equivalent ⇔the module categories Mod(A) and Mod(B) are equivalent
isomorphic algebras are Morita equivalent
Morita equivalent commutative algebras are isomorphic
Morita equivalent but non-isomorphic algebras: e.g. Matn(C) and C
Morita equivalence can be defined in arbitrary monoidal categories.
Morita classes of Frobenius algebras in fusion categories are well studied.
David Reutter Quantum graph isomorphisms March 20, 2018 16 / 23
![Page 102: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/102.jpg)
Morita theory on a slide
Morita theory: The study of algebras up to Morita equivalence.
Definition
Algebras A,B are Morita equivalent ⇔the module categories Mod(A) and Mod(B) are equivalent
isomorphic algebras are Morita equivalent
Morita equivalent commutative algebras are isomorphic
Morita equivalent but non-isomorphic algebras: e.g. Matn(C) and C
Morita equivalence can be defined in arbitrary monoidal categories.
Morita classes of Frobenius algebras in fusion categories are well studied.
David Reutter Quantum graph isomorphisms March 20, 2018 16 / 23
![Page 103: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/103.jpg)
Morita theory on a slide
Morita theory: The study of algebras up to Morita equivalence.
Definition
Algebras A,B are Morita equivalent ⇔the module categories Mod(A) and Mod(B) are equivalent
isomorphic algebras are Morita equivalent
Morita equivalent commutative algebras are isomorphic
Morita equivalent but non-isomorphic algebras: e.g. Matn(C) and C
Morita equivalence can be defined in arbitrary monoidal categories.
Morita classes of Frobenius algebras in fusion categories are well studied.
David Reutter Quantum graph isomorphisms March 20, 2018 16 / 23
![Page 104: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/104.jpg)
Morita theory on a slide
Morita theory: The study of algebras up to Morita equivalence.
Definition
Algebras A,B are Morita equivalent ⇔the module categories Mod(A) and Mod(B) are equivalent
isomorphic algebras are Morita equivalent
Morita equivalent commutative algebras are isomorphic
Morita equivalent but non-isomorphic algebras: e.g. Matn(C) and C
Morita equivalence can be defined in arbitrary monoidal categories.
Morita classes of Frobenius algebras in fusion categories are well studied.
David Reutter Quantum graph isomorphisms March 20, 2018 16 / 23
![Page 105: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/105.jpg)
Morita theory on a slide
Morita theory: The study of algebras up to Morita equivalence.
Definition
Algebras A,B are Morita equivalent ⇔the module categories Mod(A) and Mod(B) are equivalent
isomorphic algebras are Morita equivalent
Morita equivalent commutative algebras are isomorphic
Morita equivalent but non-isomorphic algebras: e.g. Matn(C) and C
Morita equivalence can be defined in arbitrary monoidal categories.
Morita classes of Frobenius algebras in fusion categories are well studied.
David Reutter Quantum graph isomorphisms March 20, 2018 16 / 23
![Page 106: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/106.jpg)
Pseudo-telepathy and QGraph
Quantum pseudo-telepathy QGraph
graph isomorphism Γ ∼= Γ′ equivalence Γ −→ Γ′
quantum graph isomorphism Γ ∼=Q Γ′ dualizable 1-morphism Γ −→ Γ′
Dualizable AF−→ B in a †-2-cat. B
†-Frobenius algebra FF in B(A,A)
F
F
=
F
F
=
F
F
F
F
=
F
F
=
F
F
In nice 2-categories, we can recover duals from Frobenius algebras.More precisely: The equivalence class of the object B in B can berecovered from the Morita class of the algebra FF .
Use this to classify graphs quantum isomorphic to a given graph.
David Reutter Quantum graph isomorphisms March 20, 2018 17 / 23
![Page 107: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/107.jpg)
Pseudo-telepathy and QGraph
Quantum pseudo-telepathy QGraph
graph isomorphism Γ ∼= Γ′ equivalence Γ −→ Γ′
quantum graph isomorphism Γ ∼=Q Γ′ dualizable 1-morphism Γ −→ Γ′
Dualizable AF−→ B in a †-2-cat. B
†-Frobenius algebra FF in B(A,A)
F
F
=
F
F
=
F
F
F
F
=
F
F
=
F
F
In nice 2-categories, we can recover duals from Frobenius algebras.More precisely: The equivalence class of the object B in B can berecovered from the Morita class of the algebra FF .
Use this to classify graphs quantum isomorphic to a given graph.
David Reutter Quantum graph isomorphisms March 20, 2018 17 / 23
![Page 108: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/108.jpg)
Pseudo-telepathy and QGraph
Quantum pseudo-telepathy QGraph
graph isomorphism Γ ∼= Γ′ equivalence Γ −→ Γ′
quantum graph isomorphism Γ ∼=Q Γ′ dualizable 1-morphism Γ −→ Γ′
Dualizable AF−→ B in a †-2-cat. B
†-Frobenius algebra FF in B(A,A)
F
F
=
F
F
=
F
F
F
F
=
F
F
=
F
F
In nice 2-categories, we can recover duals from Frobenius algebras.More precisely: The equivalence class of the object B in B can berecovered from the Morita class of the algebra FF .
Use this to classify graphs quantum isomorphic to a given graph.
David Reutter Quantum graph isomorphisms March 20, 2018 17 / 23
![Page 109: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/109.jpg)
Pseudo-telepathy and QGraph
Quantum pseudo-telepathy QGraph
graph isomorphism Γ ∼= Γ′ equivalence Γ −→ Γ′
quantum graph isomorphism Γ ∼=Q Γ′ dualizable 1-morphism Γ −→ Γ′
Dualizable AF−→ B in a †-2-cat. B
†-Frobenius algebra FF in B(A,A)
F
F
=
F
F
=
F
F
F
F
=
F
F
=
F
F
In nice 2-categories, we can recover duals from Frobenius algebras.More precisely: The equivalence class of the object B in B can berecovered from the Morita class of the algebra FF .
Use this to classify graphs quantum isomorphic to a given graph.
David Reutter Quantum graph isomorphisms March 20, 2018 17 / 23
![Page 110: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/110.jpg)
Pseudo-telepathy and QGraph
Quantum pseudo-telepathy QGraph
graph isomorphism Γ ∼= Γ′ equivalence Γ −→ Γ′
quantum graph isomorphism Γ ∼=Q Γ′ dualizable 1-morphism Γ −→ Γ′
Dualizable AF−→ B in a †-2-cat. B
†-Frobenius algebra FF in B(A,A)
F
F
=
F
F
=
F
F
F
F
=
F
F
=
F
F
In nice 2-categories, we can recover duals from Frobenius algebras.More precisely: The equivalence class of the object B in B can berecovered from the Morita class of the algebra FF .
Use this to classify graphs quantum isomorphic to a given graph.
David Reutter Quantum graph isomorphisms March 20, 2018 17 / 23
![Page 111: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/111.jpg)
Pseudo-telepathy and QGraph
Quantum pseudo-telepathy QGraph
graph isomorphism Γ ∼= Γ′ equivalence Γ −→ Γ′
quantum graph isomorphism Γ ∼=Q Γ′ dualizable 1-morphism Γ −→ Γ′
Dualizable AF−→ B in a †-2-cat. B †-Frobenius algebra FF in B(A,A)
F
F
=
F
F
=
F
F
F
F
=
F
F
=
F
F
In nice 2-categories, we can recover duals from Frobenius algebras.More precisely: The equivalence class of the object B in B can berecovered from the Morita class of the algebra FF .
Use this to classify graphs quantum isomorphic to a given graph.
David Reutter Quantum graph isomorphisms March 20, 2018 17 / 23
![Page 112: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/112.jpg)
Pseudo-telepathy and QGraph
Quantum pseudo-telepathy QGraph
graph isomorphism Γ ∼= Γ′ equivalence Γ −→ Γ′
quantum graph isomorphism Γ ∼=Q Γ′ dualizable 1-morphism Γ −→ Γ′
Dualizable AF−→ B in a †-2-cat. B †-Frobenius algebra FF in B(A,A)
F
F
=
F
F
=
F
F
F
F
=
F
F
=
F
F
In nice 2-categories, we can recover duals from Frobenius algebras.
More precisely: The equivalence class of the object B in B can berecovered from the Morita class of the algebra FF .
Use this to classify graphs quantum isomorphic to a given graph.
David Reutter Quantum graph isomorphisms March 20, 2018 17 / 23
![Page 113: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/113.jpg)
Pseudo-telepathy and QGraph
Quantum pseudo-telepathy QGraph
graph isomorphism Γ ∼= Γ′ equivalence Γ −→ Γ′
quantum graph isomorphism Γ ∼=Q Γ′ dualizable 1-morphism Γ −→ Γ′
Dualizable AF−→ B in a †-2-cat. B †-Frobenius algebra FF in B(A,A)
F
F
=
F
F
=
F
F
F
F
=
F
F
=
F
F
In nice 2-categories, we can recover duals from Frobenius algebras.More precisely: The equivalence class of the object B in B can berecovered from the Morita class of the algebra FF .
Use this to classify graphs quantum isomorphic to a given graph.
David Reutter Quantum graph isomorphisms March 20, 2018 17 / 23
![Page 114: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/114.jpg)
Pseudo-telepathy and QGraph
Quantum pseudo-telepathy QGraph
graph isomorphism Γ ∼= Γ′ equivalence Γ −→ Γ′
quantum graph isomorphism Γ ∼=Q Γ′ dualizable 1-morphism Γ −→ Γ′
Dualizable AF−→ B in QGraph †-Frobenius algebra FF in QAut(A)
F
F
=
F
F
=
F
F
F
F
=
F
F
=
F
F
In nice 2-categories, we can recover duals from Frobenius algebras.More precisely: The equivalence class of the object B in B can berecovered from the Morita class of the algebra FF .
Use this to classify graphs quantum isomorphic to a given graph.
David Reutter Quantum graph isomorphisms March 20, 2018 17 / 23
![Page 115: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/115.jpg)
Classifying quantum isomorphic graphs
There is a monoidal forgetful functor F : QAut(G ) −→ Hilb:
H
H VG
VG
P 7→ H
Definition:A dagger Frobenius algebra A in QAut(G ) is simple if F (A) ∼= End(H) forsome Hilbert space H.
Theorem
For a graph G , there is a bijective correspondence between:
isomorphism classes of graphs H quantum isomorphic to G
Morita classes of simple dagger Frobenius algebras in QAut(G )fulfilling a certain commutativity condition
drop commutativity condition ! classify quantum graphs [1,2]
David Reutter Quantum graph isomorphisms March 20, 2018 18 / 23
![Page 116: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/116.jpg)
Classifying quantum isomorphic graphs
There is a monoidal forgetful functor F : QAut(G ) −→ Hilb:
H
H VG
VG
P 7→ H
Definition:A dagger Frobenius algebra A in QAut(G ) is simple if F (A) ∼= End(H) forsome Hilbert space H.
Theorem
For a graph G , there is a bijective correspondence between:
isomorphism classes of graphs H quantum isomorphic to G
Morita classes of simple dagger Frobenius algebras in QAut(G )fulfilling a certain commutativity condition
drop commutativity condition ! classify quantum graphs [1,2]
David Reutter Quantum graph isomorphisms March 20, 2018 18 / 23
![Page 117: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/117.jpg)
Classifying quantum isomorphic graphs
There is a monoidal forgetful functor F : QAut(G ) −→ Hilb:
H
H VG
VG
P 7→ H
Definition:A dagger Frobenius algebra A in QAut(G ) is simple if F (A) ∼= End(H) forsome Hilbert space H.
Theorem
For a graph G , there is a bijective correspondence between:
isomorphism classes of graphs H quantum isomorphic to G
Morita classes of simple dagger Frobenius algebras in QAut(G )fulfilling a certain commutativity condition
drop commutativity condition ! classify quantum graphs [1,2]
David Reutter Quantum graph isomorphisms March 20, 2018 18 / 23
![Page 118: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/118.jpg)
Classifying quantum isomorphic graphs
There is a monoidal forgetful functor F : QAut(G ) −→ Hilb:
H
H VG
VG
P 7→ H
Definition:A dagger Frobenius algebra A in QAut(G ) is simple if F (A) ∼= End(H) forsome Hilbert space H.
Theorem
For a graph G , there is a bijective correspondence between:
isomorphism classes of graphs H quantum isomorphic to G
Morita classes of simple dagger Frobenius algebras in QAut(G )fulfilling a certain commutativity condition
drop commutativity condition ! classify quantum graphs [1,2]
David Reutter Quantum graph isomorphisms March 20, 2018 18 / 23
![Page 119: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/119.jpg)
Classifying quantum isomorphic graphs
There is a monoidal forgetful functor F : QAut(G ) −→ Hilb:
H
H VG
VG
P 7→ H
Definition:A dagger Frobenius algebra A in QAut(G ) is simple if F (A) ∼= End(H) forsome Hilbert space H.
Theorem
For a quantum graph G , there is a bijective correspondence between:
isomorphism classes of quantum graphs H quantum isomorphic to G
Morita classes of simple dagger Frobenius algebras in QAut(G )
fulfilling a certain commutativity condition
drop commutativity condition ! classify quantum graphs [1,2]
David Reutter Quantum graph isomorphisms March 20, 2018 18 / 23
[1] Weaver — Quantum graphs as quantum relations. 2015
[2] Duan, Severini, Winter — Zero error communication [...] theta functions. 2010
![Page 120: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/120.jpg)
Frobenius algebras in classical subcategories
QAut(G ) is too large.
Let’s focus on an easier subcategory:
The classical subcategory Aut(G ) : direct sums of classical automorphisms
Definition:A group of central type is a group H with a 2-cocycle ψ : H × H −→ U(1)such that CHψ is a simple algebra.Example:The Pauli matrices make the group Z2 × Z2 into a group of central type:
C (Z2 × Z2)ψ −→ End(C2) (a, b) 7→ X aZb
Theorem
Morita classes of simple dagger Frobenius algebras in Aut(G ) are inbijective correspondence with central type subgroups of Aut(G ).
What about the commutativity condition?
David Reutter Quantum graph isomorphisms March 20, 2018 19 / 23
![Page 121: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/121.jpg)
Frobenius algebras in classical subcategories
QAut(G ) is too large. Let’s focus on an easier subcategory:
The classical subcategory Aut(G ) : direct sums of classical automorphisms
Definition:A group of central type is a group H with a 2-cocycle ψ : H × H −→ U(1)such that CHψ is a simple algebra.Example:The Pauli matrices make the group Z2 × Z2 into a group of central type:
C (Z2 × Z2)ψ −→ End(C2) (a, b) 7→ X aZb
Theorem
Morita classes of simple dagger Frobenius algebras in Aut(G ) are inbijective correspondence with central type subgroups of Aut(G ).
What about the commutativity condition?
David Reutter Quantum graph isomorphisms March 20, 2018 19 / 23
![Page 122: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/122.jpg)
Frobenius algebras in classical subcategories
QAut(G ) is too large. Let’s focus on an easier subcategory:
The classical subcategory Aut(G ) : direct sums of classical automorphisms
Definition:A group of central type is a group H with a 2-cocycle ψ : H × H −→ U(1)such that CHψ is a simple algebra.
Example:The Pauli matrices make the group Z2 × Z2 into a group of central type:
C (Z2 × Z2)ψ −→ End(C2) (a, b) 7→ X aZb
Theorem
Morita classes of simple dagger Frobenius algebras in Aut(G ) are inbijective correspondence with central type subgroups of Aut(G ).
What about the commutativity condition?
David Reutter Quantum graph isomorphisms March 20, 2018 19 / 23
![Page 123: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/123.jpg)
Frobenius algebras in classical subcategories
QAut(G ) is too large. Let’s focus on an easier subcategory:
The classical subcategory Aut(G ) : direct sums of classical automorphisms
Definition:A group of central type is a group H with a 2-cocycle ψ : H × H −→ U(1)such that CHψ is a simple algebra.Example:The Pauli matrices make the group Z2 × Z2 into a group of central type:
C (Z2 × Z2)ψ −→ End(C2) (a, b) 7→ X aZb
Theorem
Morita classes of simple dagger Frobenius algebras in Aut(G ) are inbijective correspondence with central type subgroups of Aut(G ).
What about the commutativity condition?
David Reutter Quantum graph isomorphisms March 20, 2018 19 / 23
![Page 124: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/124.jpg)
Frobenius algebras in classical subcategories
QAut(G ) is too large. Let’s focus on an easier subcategory:
The classical subcategory Aut(G ) : direct sums of classical automorphisms
Definition:A group of central type is a group H with a 2-cocycle ψ : H × H −→ U(1)such that CHψ is a simple algebra.Example:The Pauli matrices make the group Z2 × Z2 into a group of central type:
C (Z2 × Z2)ψ −→ End(C2) (a, b) 7→ X aZb
Theorem
Morita classes of simple dagger Frobenius algebras in Aut(G ) are inbijective correspondence with central type subgroups of Aut(G ).
What about the commutativity condition?
David Reutter Quantum graph isomorphisms March 20, 2018 19 / 23
![Page 125: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/125.jpg)
Frobenius algebras in classical subcategories
QAut(G ) is too large. Let’s focus on an easier subcategory:
The classical subcategory Aut(G ) : direct sums of classical automorphisms
Definition:A group of central type is a group H with a 2-cocycle ψ : H × H −→ U(1)such that CHψ is a simple algebra.Example:The Pauli matrices make the group Z2 × Z2 into a group of central type:
C (Z2 × Z2)ψ −→ End(C2) (a, b) 7→ X aZb
Theorem
Morita classes of simple dagger Frobenius algebras in Aut(G ) are inbijective correspondence with central type subgroups of Aut(G ).
What about the commutativity condition?
David Reutter Quantum graph isomorphisms March 20, 2018 19 / 23
![Page 126: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/126.jpg)
Coisotropic stabilizers
Let H be a group of central type with 2-cocycle ψ.Define ρ(a, b) := ψ(a, b)ψ(b, a). ρ is a symplectic form on H.
The orthogonal complement of a subgroup S ⊆ H is
S⊥ = {h ∈ H | ρ(s, h) = 1 ∀s ∈ S ∩ Zh}
Zh = {s ∈ H | sh = hs}
A subgroup S ⊆ H is coisotropic if S⊥ ⊆ S .Let G be a graph. A subgroup H ⊆ Aut(G ) has coisotropic stabilizers ifStab(v) ∩ H is coisotropic for all vertices v of G .
Theorem
Let H be a central type subgroup of Aut(G ). The corresponding simple
dagger Frobenius algebra in Aut(G ) fulfills the commutativity condition ifand only if H has coisotropic stabilizers.
Given: A central type subgroup of Aut(G ) with coisotropic stabilizers.Get: A graph G ′ quantum isomorphic to G .If G has no quantum symmetries: get all quantum isomorphic graphs G ′
David Reutter Quantum graph isomorphisms March 20, 2018 20 / 23
![Page 127: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/127.jpg)
Coisotropic stabilizers
Let H be a group of central type with 2-cocycle ψ.Define ρ(a, b) := ψ(a, b)ψ(b, a). ρ is a symplectic form on H.The orthogonal complement of a subgroup S ⊆ H is
S⊥ = {h ∈ H | ρ(s, h) = 1 ∀s ∈ S ∩ Zh}
Zh = {s ∈ H | sh = hs}
A subgroup S ⊆ H is coisotropic if S⊥ ⊆ S .Let G be a graph. A subgroup H ⊆ Aut(G ) has coisotropic stabilizers ifStab(v) ∩ H is coisotropic for all vertices v of G .
Theorem
Let H be a central type subgroup of Aut(G ). The corresponding simple
dagger Frobenius algebra in Aut(G ) fulfills the commutativity condition ifand only if H has coisotropic stabilizers.
Given: A central type subgroup of Aut(G ) with coisotropic stabilizers.Get: A graph G ′ quantum isomorphic to G .If G has no quantum symmetries: get all quantum isomorphic graphs G ′
David Reutter Quantum graph isomorphisms March 20, 2018 20 / 23
![Page 128: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/128.jpg)
Coisotropic stabilizers
Let H be a group of central type with 2-cocycle ψ.Define ρ(a, b) := ψ(a, b)ψ(b, a). ρ is a symplectic form on H.The orthogonal complement of a subgroup S ⊆ H is
S⊥ = {h ∈ H | ρ(s, h) = 1 ∀s ∈ S ∩ Zh}
Zh = {s ∈ H | sh = hs}
A subgroup S ⊆ H is coisotropic if S⊥ ⊆ S .Let G be a graph. A subgroup H ⊆ Aut(G ) has coisotropic stabilizers ifStab(v) ∩ H is coisotropic for all vertices v of G .
Theorem
Let H be a central type subgroup of Aut(G ). The corresponding simple
dagger Frobenius algebra in Aut(G ) fulfills the commutativity condition ifand only if H has coisotropic stabilizers.
Given: A central type subgroup of Aut(G ) with coisotropic stabilizers.Get: A graph G ′ quantum isomorphic to G .If G has no quantum symmetries: get all quantum isomorphic graphs G ′
David Reutter Quantum graph isomorphisms March 20, 2018 20 / 23
![Page 129: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/129.jpg)
Coisotropic stabilizers
Let H be a group of central type with 2-cocycle ψ.Define ρ(a, b) := ψ(a, b)ψ(b, a). ρ is a symplectic form on H.The orthogonal complement of a subgroup S ⊆ H is
S⊥ = {h ∈ H | ρ(s, h) = 1 ∀s ∈ S ∩ Zh}
Zh = {s ∈ H | sh = hs}
A subgroup S ⊆ H is coisotropic if S⊥ ⊆ S .
Let G be a graph. A subgroup H ⊆ Aut(G ) has coisotropic stabilizers ifStab(v) ∩ H is coisotropic for all vertices v of G .
Theorem
Let H be a central type subgroup of Aut(G ). The corresponding simple
dagger Frobenius algebra in Aut(G ) fulfills the commutativity condition ifand only if H has coisotropic stabilizers.
Given: A central type subgroup of Aut(G ) with coisotropic stabilizers.Get: A graph G ′ quantum isomorphic to G .If G has no quantum symmetries: get all quantum isomorphic graphs G ′
David Reutter Quantum graph isomorphisms March 20, 2018 20 / 23
![Page 130: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/130.jpg)
Coisotropic stabilizers
Let H be a group of central type with 2-cocycle ψ.Define ρ(a, b) := ψ(a, b)ψ(b, a). ρ is a symplectic form on H.The orthogonal complement of a subgroup S ⊆ H is
S⊥ = {h ∈ H | ρ(s, h) = 1 ∀s ∈ S ∩ Zh}
Zh = {s ∈ H | sh = hs}
A subgroup S ⊆ H is coisotropic if S⊥ ⊆ S .Let G be a graph. A subgroup H ⊆ Aut(G ) has coisotropic stabilizers ifStab(v) ∩ H is coisotropic for all vertices v of G .
Theorem
Let H be a central type subgroup of Aut(G ). The corresponding simple
dagger Frobenius algebra in Aut(G ) fulfills the commutativity condition ifand only if H has coisotropic stabilizers.
Given: A central type subgroup of Aut(G ) with coisotropic stabilizers.Get: A graph G ′ quantum isomorphic to G .If G has no quantum symmetries: get all quantum isomorphic graphs G ′
David Reutter Quantum graph isomorphisms March 20, 2018 20 / 23
![Page 131: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/131.jpg)
Coisotropic stabilizers
Let H be a group of central type with 2-cocycle ψ.Define ρ(a, b) := ψ(a, b)ψ(b, a). ρ is a symplectic form on H.The orthogonal complement of a subgroup S ⊆ H is
S⊥ = {h ∈ H | ρ(s, h) = 1 ∀s ∈ S ∩ Zh}
Zh = {s ∈ H | sh = hs}
A subgroup S ⊆ H is coisotropic if S⊥ ⊆ S .Let G be a graph. A subgroup H ⊆ Aut(G ) has coisotropic stabilizers ifStab(v) ∩ H is coisotropic for all vertices v of G .
Theorem
Let H be a central type subgroup of Aut(G ). The corresponding simple
dagger Frobenius algebra in Aut(G ) fulfills the commutativity condition ifand only if H has coisotropic stabilizers.
Given: A central type subgroup of Aut(G ) with coisotropic stabilizers.Get: A graph G ′ quantum isomorphic to G .If G has no quantum symmetries: get all quantum isomorphic graphs G ′
David Reutter Quantum graph isomorphisms March 20, 2018 20 / 23
![Page 132: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/132.jpg)
Coisotropic stabilizers
Let H be a group of central type with 2-cocycle ψ.Define ρ(a, b) := ψ(a, b)ψ(b, a). ρ is a symplectic form on H.The orthogonal complement of a subgroup S ⊆ H is
S⊥ = {h ∈ H | ρ(s, h) = 1 ∀s ∈ S ∩ Zh}
Zh = {s ∈ H | sh = hs}
A subgroup S ⊆ H is coisotropic if S⊥ ⊆ S .Let G be a graph. A subgroup H ⊆ Aut(G ) has coisotropic stabilizers ifStab(v) ∩ H is coisotropic for all vertices v of G .
Theorem
Let H be a central type subgroup of Aut(G ). The corresponding simple
dagger Frobenius algebra in Aut(G ) fulfills the commutativity condition ifand only if H has coisotropic stabilizers.
Given: A central type subgroup of Aut(G ) with coisotropic stabilizers.
Get: A graph G ′ quantum isomorphic to G .If G has no quantum symmetries: get all quantum isomorphic graphs G ′
David Reutter Quantum graph isomorphisms March 20, 2018 20 / 23
![Page 133: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/133.jpg)
Coisotropic stabilizers
Let H be a group of central type with 2-cocycle ψ.Define ρ(a, b) := ψ(a, b)ψ(b, a). ρ is a symplectic form on H.The orthogonal complement of a subgroup S ⊆ H is
S⊥ = {h ∈ H | ρ(s, h) = 1 ∀s ∈ S ∩ Zh}
Zh = {s ∈ H | sh = hs}
A subgroup S ⊆ H is coisotropic if S⊥ ⊆ S .Let G be a graph. A subgroup H ⊆ Aut(G ) has coisotropic stabilizers ifStab(v) ∩ H is coisotropic for all vertices v of G .
Theorem
Let H be a central type subgroup of Aut(G ). The corresponding simple
dagger Frobenius algebra in Aut(G ) fulfills the commutativity condition ifand only if H has coisotropic stabilizers.
Given: A central type subgroup of Aut(G ) with coisotropic stabilizers.Get: A graph G ′ quantum isomorphic to G .
If G has no quantum symmetries: get all quantum isomorphic graphs G ′
David Reutter Quantum graph isomorphisms March 20, 2018 20 / 23
![Page 134: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/134.jpg)
Coisotropic stabilizers
Let H be a group of central type with 2-cocycle ψ.Define ρ(a, b) := ψ(a, b)ψ(b, a). ρ is a symplectic form on H.The orthogonal complement of a subgroup S ⊆ H is
S⊥ = {h ∈ H | ρ(s, h) = 1 ∀s ∈ S ∩ Zh}
Zh = {s ∈ H | sh = hs}
A subgroup S ⊆ H is coisotropic if S⊥ ⊆ S .Let G be a graph. A subgroup H ⊆ Aut(G ) has coisotropic stabilizers ifStab(v) ∩ H is coisotropic for all vertices v of G .
Theorem
Let H be a central type subgroup of Aut(G ). The corresponding simple
dagger Frobenius algebra in Aut(G ) fulfills the commutativity condition ifand only if H has coisotropic stabilizers.
Given: A central type subgroup of Aut(G ) with coisotropic stabilizers.Get: A graph G ′ quantum isomorphic to G .If G has no quantum symmetries: get all quantum isomorphic graphs G ′
David Reutter Quantum graph isomorphisms March 20, 2018 20 / 23
![Page 135: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/135.jpg)
The construction in the abelian case
Let G be a graph with vertex set VG .Given: An abelian central type subgroup H ⊆ Aut(G ) with corresponding2-cocycle ψ which has coisotropic stabilizers.
Construct: A graph G ′ quantum isomorphic to G .
Let O ⊆ VG be an H-orbit and let Stab(O) ⊆ H be the stabilizersubgroup of this orbit.
Let GO be the graph G restricted to the orbit O.
Consider the disjoint union graph tOGO .
For every orbit O pick a 1-cochain φO on Stab(O) such thatdφO = ψ|Stab(O).
For every pair of orbits O and O ′, consider the 1-cocycle φOφO′ onStab(O) ∩ Stab(O ′). This extends to a 1-cocycle on the group H ofthe form ρ(hO,O′ ,−) for some hO,O′ ∈ H.
Reconnect the disjoint components of tOGO as follows:
v ∈ O ∼G ′ w ∈ O ′ ⇔ hO,Ov ∼G w
David Reutter Quantum graph isomorphisms March 20, 2018 21 / 23
![Page 136: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/136.jpg)
The construction in the abelian case
Let G be a graph with vertex set VG .Given: An abelian central type subgroup H ⊆ Aut(G ) with corresponding2-cocycle ψ which has coisotropic stabilizers.Construct: A graph G ′ quantum isomorphic to G .
Let O ⊆ VG be an H-orbit and let Stab(O) ⊆ H be the stabilizersubgroup of this orbit.
Let GO be the graph G restricted to the orbit O.
Consider the disjoint union graph tOGO .
For every orbit O pick a 1-cochain φO on Stab(O) such thatdφO = ψ|Stab(O).
For every pair of orbits O and O ′, consider the 1-cocycle φOφO′ onStab(O) ∩ Stab(O ′). This extends to a 1-cocycle on the group H ofthe form ρ(hO,O′ ,−) for some hO,O′ ∈ H.
Reconnect the disjoint components of tOGO as follows:
v ∈ O ∼G ′ w ∈ O ′ ⇔ hO,Ov ∼G w
David Reutter Quantum graph isomorphisms March 20, 2018 21 / 23
![Page 137: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/137.jpg)
The construction in the abelian case
Let G be a graph with vertex set VG .Given: An abelian central type subgroup H ⊆ Aut(G ) with corresponding2-cocycle ψ which has coisotropic stabilizers.Construct: A graph G ′ quantum isomorphic to G .
Let O ⊆ VG be an H-orbit and let Stab(O) ⊆ H be the stabilizersubgroup of this orbit.
Let GO be the graph G restricted to the orbit O.
Consider the disjoint union graph tOGO .
For every orbit O pick a 1-cochain φO on Stab(O) such thatdφO = ψ|Stab(O).
For every pair of orbits O and O ′, consider the 1-cocycle φOφO′ onStab(O) ∩ Stab(O ′). This extends to a 1-cocycle on the group H ofthe form ρ(hO,O′ ,−) for some hO,O′ ∈ H.
Reconnect the disjoint components of tOGO as follows:
v ∈ O ∼G ′ w ∈ O ′ ⇔ hO,Ov ∼G w
David Reutter Quantum graph isomorphisms March 20, 2018 21 / 23
![Page 138: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/138.jpg)
The construction in the abelian case
Let G be a graph with vertex set VG .Given: An abelian central type subgroup H ⊆ Aut(G ) with corresponding2-cocycle ψ which has coisotropic stabilizers.Construct: A graph G ′ quantum isomorphic to G .
Let O ⊆ VG be an H-orbit and let Stab(O) ⊆ H be the stabilizersubgroup of this orbit.
Let GO be the graph G restricted to the orbit O.
Consider the disjoint union graph tOGO .
For every orbit O pick a 1-cochain φO on Stab(O) such thatdφO = ψ|Stab(O).
For every pair of orbits O and O ′, consider the 1-cocycle φOφO′ onStab(O) ∩ Stab(O ′). This extends to a 1-cocycle on the group H ofthe form ρ(hO,O′ ,−) for some hO,O′ ∈ H.
Reconnect the disjoint components of tOGO as follows:
v ∈ O ∼G ′ w ∈ O ′ ⇔ hO,Ov ∼G w
David Reutter Quantum graph isomorphisms March 20, 2018 21 / 23
![Page 139: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/139.jpg)
The construction in the abelian case
Let G be a graph with vertex set VG .Given: An abelian central type subgroup H ⊆ Aut(G ) with corresponding2-cocycle ψ which has coisotropic stabilizers.Construct: A graph G ′ quantum isomorphic to G .
Let O ⊆ VG be an H-orbit and let Stab(O) ⊆ H be the stabilizersubgroup of this orbit.
Let GO be the graph G restricted to the orbit O.
Consider the disjoint union graph tOGO .
For every orbit O pick a 1-cochain φO on Stab(O) such thatdφO = ψ|Stab(O).
For every pair of orbits O and O ′, consider the 1-cocycle φOφO′ onStab(O) ∩ Stab(O ′). This extends to a 1-cocycle on the group H ofthe form ρ(hO,O′ ,−) for some hO,O′ ∈ H.
Reconnect the disjoint components of tOGO as follows:
v ∈ O ∼G ′ w ∈ O ′ ⇔ hO,Ov ∼G w
David Reutter Quantum graph isomorphisms March 20, 2018 21 / 23
![Page 140: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/140.jpg)
The construction in the abelian case
Let G be a graph with vertex set VG .Given: An abelian central type subgroup H ⊆ Aut(G ) with corresponding2-cocycle ψ which has coisotropic stabilizers.Construct: A graph G ′ quantum isomorphic to G .
Let O ⊆ VG be an H-orbit and let Stab(O) ⊆ H be the stabilizersubgroup of this orbit.
Let GO be the graph G restricted to the orbit O.
Consider the disjoint union graph tOGO .
For every orbit O pick a 1-cochain φO on Stab(O) such thatdφO = ψ|Stab(O).
For every pair of orbits O and O ′, consider the 1-cocycle φOφO′ onStab(O) ∩ Stab(O ′). This extends to a 1-cocycle on the group H ofthe form ρ(hO,O′ ,−) for some hO,O′ ∈ H.
Reconnect the disjoint components of tOGO as follows:
v ∈ O ∼G ′ w ∈ O ′ ⇔ hO,Ov ∼G w
David Reutter Quantum graph isomorphisms March 20, 2018 21 / 23
![Page 141: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/141.jpg)
The construction in the abelian case
Let G be a graph with vertex set VG .Given: An abelian central type subgroup H ⊆ Aut(G ) with corresponding2-cocycle ψ which has coisotropic stabilizers.Construct: A graph G ′ quantum isomorphic to G .
Let O ⊆ VG be an H-orbit and let Stab(O) ⊆ H be the stabilizersubgroup of this orbit.
Let GO be the graph G restricted to the orbit O.
Consider the disjoint union graph tOGO .
For every orbit O pick a 1-cochain φO on Stab(O) such thatdφO = ψ|Stab(O).
For every pair of orbits O and O ′, consider the 1-cocycle φOφO′ onStab(O) ∩ Stab(O ′). This extends to a 1-cocycle on the group H ofthe form ρ(hO,O′ ,−) for some hO,O′ ∈ H.
Reconnect the disjoint components of tOGO as follows:
v ∈ O ∼G ′ w ∈ O ′ ⇔ hO,Ov ∼G w
David Reutter Quantum graph isomorphisms March 20, 2018 21 / 23
![Page 142: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/142.jpg)
The construction in the abelian case
Let G be a graph with vertex set VG .Given: An abelian central type subgroup H ⊆ Aut(G ) with corresponding2-cocycle ψ which has coisotropic stabilizers.Construct: A graph G ′ quantum isomorphic to G .
Let O ⊆ VG be an H-orbit and let Stab(O) ⊆ H be the stabilizersubgroup of this orbit.
Let GO be the graph G restricted to the orbit O.
Consider the disjoint union graph tOGO .
For every orbit O pick a 1-cochain φO on Stab(O) such thatdφO = ψ|Stab(O).
For every pair of orbits O and O ′, consider the 1-cocycle φOφO′ onStab(O) ∩ Stab(O ′). This extends to a 1-cocycle on the group H ofthe form ρ(hO,O′ ,−) for some hO,O′ ∈ H.
Reconnect the disjoint components of tOGO as follows:
v ∈ O ∼G ′ w ∈ O ′ ⇔ hO,Ov ∼G w
David Reutter Quantum graph isomorphisms March 20, 2018 21 / 23
![Page 143: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/143.jpg)
An example: binary magic squares (BMS)
BMS: a 3× 3 square with
{entries in {0, 1}
rows and columns add up to 0 mod 2
Define a graph ΓBMS :
vertices: partial BMS — only one row or column filled — 24 vertices
0
0
101
7 7
7 7
7 7
7 7
7 7
7 7
7 7
edge between two vertices if the partial BMS contradict each other
Bit-flip symmetries
of this graph
form a subgroup (Z2)4 ≤ Aut(ΓBMS).
generators:7 7 ·7 7 ·· · ·
· 7 7
· 7 7
· · ·
· · ·7 7 ·7 7 ·
· · ·· 7 7
· 7 7
⇒ (Z2)4 is a group of central type with coisotropic stabilizers
⇒ classification gives graph Γ′ quantum isomorphic to Γ
⇒ Γ′ coincides with a graph in [1] coming from the Mermin-Peres square
⇒ Pseudo-telepathy from the symmetries of classical magic squares
David Reutter Quantum graph isomorphisms March 20, 2018 22 / 23
[1] Atserias, Mancinska, Roberson, Samal, Severini and Varvitsiotis — Quantum [...] graph isomorphisms. 2017
![Page 144: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/144.jpg)
An example: binary magic squares (BMS)
BMS: a 3× 3 square with
{entries in {0, 1}
rows and columns add up to 0 mod 2
Define a graph ΓBMS :
vertices: partial BMS — only one row or column filled — 24 vertices
0 0 00 0 00 0 0
0 0 01 1 01 1 0
1 1 00 1 11 0 1
7 7
7 7
7 7
7 7
7 7
7 7
7 7
edge between two vertices if the partial BMS contradict each other
Bit-flip symmetries
of this graph
form a subgroup (Z2)4 ≤ Aut(ΓBMS).
generators:7 7 ·7 7 ·· · ·
· 7 7
· 7 7
· · ·
· · ·7 7 ·7 7 ·
· · ·· 7 7
· 7 7
⇒ (Z2)4 is a group of central type with coisotropic stabilizers
⇒ classification gives graph Γ′ quantum isomorphic to Γ
⇒ Γ′ coincides with a graph in [1] coming from the Mermin-Peres square
⇒ Pseudo-telepathy from the symmetries of classical magic squares
David Reutter Quantum graph isomorphisms March 20, 2018 22 / 23
[1] Atserias, Mancinska, Roberson, Samal, Severini and Varvitsiotis — Quantum [...] graph isomorphisms. 2017
![Page 145: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/145.jpg)
An example: binary magic squares (BMS)
BMS: a 3× 3 square with
{entries in {0, 1}
rows and columns add up to 0 mod 2Define a graph ΓBMS :
vertices: partial BMS — only one row or column filled — 24 vertices0 0 0· · ·· · ·
· · ·· · ·1 1 0
1 · ·0 · ·1 · ·
7 7
7 7
7 7
7 7
7 7
7 7
7 7
edge between two vertices if the partial BMS contradict each other
Bit-flip symmetries
of this graph
form a subgroup (Z2)4 ≤ Aut(ΓBMS).
generators:7 7 ·7 7 ·· · ·
· 7 7
· 7 7
· · ·
· · ·7 7 ·7 7 ·
· · ·· 7 7
· 7 7
⇒ (Z2)4 is a group of central type with coisotropic stabilizers
⇒ classification gives graph Γ′ quantum isomorphic to Γ
⇒ Γ′ coincides with a graph in [1] coming from the Mermin-Peres square
⇒ Pseudo-telepathy from the symmetries of classical magic squares
David Reutter Quantum graph isomorphisms March 20, 2018 22 / 23
[1] Atserias, Mancinska, Roberson, Samal, Severini and Varvitsiotis — Quantum [...] graph isomorphisms. 2017
![Page 146: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/146.jpg)
An example: binary magic squares (BMS)
BMS: a 3× 3 square with
{entries in {0, 1}
rows and columns add up to 0 mod 2Define a graph ΓBMS :
vertices: partial BMS — only one row or column filled — 24 vertices0 0 0· · ·· · ·
· · ·· · ·1 1 0
1 · ·0 · ·1 · ·
7 7
7 7
7 7
7 7
7 7
7 7
7 7
edge between two vertices if the partial BMS contradict each other
Bit-flip symmetries
of this graph
form a subgroup (Z2)4 ≤ Aut(ΓBMS).
generators:7 7 ·7 7 ·· · ·
· 7 7
· 7 7
· · ·
· · ·7 7 ·7 7 ·
· · ·· 7 7
· 7 7
⇒ (Z2)4 is a group of central type with coisotropic stabilizers
⇒ classification gives graph Γ′ quantum isomorphic to Γ
⇒ Γ′ coincides with a graph in [1] coming from the Mermin-Peres square
⇒ Pseudo-telepathy from the symmetries of classical magic squares
David Reutter Quantum graph isomorphisms March 20, 2018 22 / 23
[1] Atserias, Mancinska, Roberson, Samal, Severini and Varvitsiotis — Quantum [...] graph isomorphisms. 2017
![Page 147: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/147.jpg)
An example: binary magic squares (BMS)
BMS: a 3× 3 square with
{entries in {0, 1}
rows and columns add up to 0 mod 2Define a graph ΓBMS :
vertices: partial BMS — only one row or column filled — 24 vertices0 0 00 0 00 0 0
0 0 01 1 01 1 0
1 1 00 1 11 0 1
7 7
7 7
7 7
7 7
7 7
7 7
7 7
edge between two vertices if the partial BMS contradict each other
Bit-flip symmetries
of this graph form a subgroup (Z2)4 ≤ Aut(ΓBMS).
generators:7 7 ·7 7 ·· · ·
· 7 7
· 7 7
· · ·
· · ·7 7 ·7 7 ·
· · ·· 7 7
· 7 7
⇒ (Z2)4 is a group of central type with coisotropic stabilizers
⇒ classification gives graph Γ′ quantum isomorphic to Γ
⇒ Γ′ coincides with a graph in [1] coming from the Mermin-Peres square
⇒ Pseudo-telepathy from the symmetries of classical magic squares
David Reutter Quantum graph isomorphisms March 20, 2018 22 / 23
[1] Atserias, Mancinska, Roberson, Samal, Severini and Varvitsiotis — Quantum [...] graph isomorphisms. 2017
![Page 148: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/148.jpg)
An example: binary magic squares (BMS)
BMS: a 3× 3 square with
{entries in {0, 1}
rows and columns add up to 0 mod 2Define a graph ΓBMS :
vertices: partial BMS — only one row or column filled — 24 vertices0 0 00 0 00 0 0
0 0 01 1 01 1 0
1 1 00 1 11 0 1
7 7
7 7
7 7
7 7
7 7
7 7
7 7
edge between two vertices if the partial BMS contradict each other
Bit-flip symmetries
of this graph form a subgroup (Z2)4 ≤ Aut(ΓBMS).
generators:7 7 ·7 7 ·· · ·
· 7 7
· 7 7
· · ·
· · ·7 7 ·7 7 ·
· · ·· 7 7
· 7 7
⇒ (Z2)4 is a group of central type with coisotropic stabilizers
⇒ classification gives graph Γ′ quantum isomorphic to Γ
⇒ Γ′ coincides with a graph in [1] coming from the Mermin-Peres square
⇒ Pseudo-telepathy from the symmetries of classical magic squares
David Reutter Quantum graph isomorphisms March 20, 2018 22 / 23
[1] Atserias, Mancinska, Roberson, Samal, Severini and Varvitsiotis — Quantum [...] graph isomorphisms. 2017
![Page 149: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/149.jpg)
An example: binary magic squares (BMS)
BMS: a 3× 3 square with
{entries in {0, 1}
rows and columns add up to 0 mod 2Define a graph ΓBMS :
vertices: partial BMS — only one row or column filled — 24 vertices1 1 01 1 00 0 0
0 1 10 1 10 0 0
1 0 10 1 11 1 0
7 7
7 7
7 7
7 7
7 7
7 7
7 7
edge between two vertices if the partial BMS contradict each other
Bit-flip symmetries
of this graph form a subgroup (Z2)4 ≤ Aut(ΓBMS).
generators:7 7 ·7 7 ·· · ·
· 7 7
· 7 7
· · ·
· · ·7 7 ·7 7 ·
· · ·· 7 7
· 7 7
⇒ (Z2)4 is a group of central type with coisotropic stabilizers
⇒ classification gives graph Γ′ quantum isomorphic to Γ
⇒ Γ′ coincides with a graph in [1] coming from the Mermin-Peres square
⇒ Pseudo-telepathy from the symmetries of classical magic squares
David Reutter Quantum graph isomorphisms March 20, 2018 22 / 23
[1] Atserias, Mancinska, Roberson, Samal, Severini and Varvitsiotis — Quantum [...] graph isomorphisms. 2017
![Page 150: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/150.jpg)
An example: binary magic squares (BMS)
BMS: a 3× 3 square with
{entries in {0, 1}
rows and columns add up to 0 mod 2Define a graph ΓBMS :
vertices: partial BMS — only one row or column filled — 24 vertices0 0 0· · ·· · ·
· · ·· · ·1 1 0
1 · ·0 · ·1 · ·
7 7
7 7
7 7
7 7
7 7
7 7
7 7
edge between two vertices if the partial BMS contradict each other
Bit-flip symmetries of this graph
form a subgroup (Z2)4 ≤ Aut(ΓBMS).
generators:7 7 ·7 7 ·· · ·
· 7 7
· 7 7
· · ·
· · ·7 7 ·7 7 ·
· · ·· 7 7
· 7 7
⇒ (Z2)4 is a group of central type with coisotropic stabilizers
⇒ classification gives graph Γ′ quantum isomorphic to Γ
⇒ Γ′ coincides with a graph in [1] coming from the Mermin-Peres square
⇒ Pseudo-telepathy from the symmetries of classical magic squares
David Reutter Quantum graph isomorphisms March 20, 2018 22 / 23
[1] Atserias, Mancinska, Roberson, Samal, Severini and Varvitsiotis — Quantum [...] graph isomorphisms. 2017
![Page 151: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/151.jpg)
An example: binary magic squares (BMS)
BMS: a 3× 3 square with
{entries in {0, 1}
rows and columns add up to 0 mod 2Define a graph ΓBMS :
vertices: partial BMS — only one row or column filled — 24 vertices0 0 0· · ·· · ·
· · ·· · ·1 1 0
1 · ·0 · ·1 · ·
7 7
7 7
7 7
7 7
7 7
7 7
7 7
edge between two vertices if the partial BMS contradict each other
Bit-flip symmetries of this graph
form a subgroup (Z2)4 ≤ Aut(ΓBMS).
generators:7 7 ·7 7 ·· · ·
· 7 7
· 7 7
· · ·
· · ·7 7 ·7 7 ·
· · ·· 7 7
· 7 7
⇒ (Z2)4 is a group of central type with coisotropic stabilizers
⇒ classification gives graph Γ′ quantum isomorphic to Γ
⇒ Γ′ coincides with a graph in [1] coming from the Mermin-Peres square
⇒ Pseudo-telepathy from the symmetries of classical magic squares
David Reutter Quantum graph isomorphisms March 20, 2018 22 / 23
[1] Atserias, Mancinska, Roberson, Samal, Severini and Varvitsiotis — Quantum [...] graph isomorphisms. 2017
![Page 152: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/152.jpg)
An example: binary magic squares (BMS)
BMS: a 3× 3 square with
{entries in {0, 1}
rows and columns add up to 0 mod 2Define a graph ΓBMS :
vertices: partial BMS — only one row or column filled — 24 vertices1 1 0· · ·· · ·
· · ·· · ·0 0 0
1 · ·0 · ·1 · ·
7 7
7 7
7 7
7 7
7 7
7 7
7 7
edge between two vertices if the partial BMS contradict each other
Bit-flip symmetries of this graph
form a subgroup (Z2)4 ≤ Aut(ΓBMS).
generators:7 7 ·7 7 ·· · ·
· 7 7
· 7 7
· · ·
· · ·7 7 ·7 7 ·
· · ·· 7 7
· 7 7
⇒ (Z2)4 is a group of central type with coisotropic stabilizers
⇒ classification gives graph Γ′ quantum isomorphic to Γ
⇒ Γ′ coincides with a graph in [1] coming from the Mermin-Peres square
⇒ Pseudo-telepathy from the symmetries of classical magic squares
David Reutter Quantum graph isomorphisms March 20, 2018 22 / 23
[1] Atserias, Mancinska, Roberson, Samal, Severini and Varvitsiotis — Quantum [...] graph isomorphisms. 2017
![Page 153: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/153.jpg)
An example: binary magic squares (BMS)
BMS: a 3× 3 square with
{entries in {0, 1}
rows and columns add up to 0 mod 2Define a graph ΓBMS :
vertices: partial BMS — only one row or column filled — 24 vertices1 1 0· · ·· · ·
· · ·· · ·0 0 0
1 · ·0 · ·1 · ·
7 7
7 7
7 7
7 7
7 7
7 7
7 7
edge between two vertices if the partial BMS contradict each other
Bit-flip symmetries of this graph form a subgroup (Z2)4 ≤ Aut(ΓBMS).
generators:7 7 ·7 7 ·· · ·
· 7 7
· 7 7
· · ·
· · ·7 7 ·7 7 ·
· · ·· 7 7
· 7 7
⇒ (Z2)4 is a group of central type with coisotropic stabilizers
⇒ classification gives graph Γ′ quantum isomorphic to Γ
⇒ Γ′ coincides with a graph in [1] coming from the Mermin-Peres square
⇒ Pseudo-telepathy from the symmetries of classical magic squares
David Reutter Quantum graph isomorphisms March 20, 2018 22 / 23
[1] Atserias, Mancinska, Roberson, Samal, Severini and Varvitsiotis — Quantum [...] graph isomorphisms. 2017
![Page 154: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/154.jpg)
An example: binary magic squares (BMS)
BMS: a 3× 3 square with
{entries in {0, 1}
rows and columns add up to 0 mod 2Define a graph ΓBMS :
vertices: partial BMS — only one row or column filled — 24 vertices1 1 0· · ·· · ·
· · ·· · ·0 0 0
1 · ·0 · ·1 · ·
7 7
7 7
7 7
7 7
7 7
7 7
7 7
edge between two vertices if the partial BMS contradict each other
Bit-flip symmetries of this graph form a subgroup (Z2)4 ≤ Aut(ΓBMS).
generators:7 7 ·7 7 ·· · ·
· 7 7
· 7 7
· · ·
· · ·7 7 ·7 7 ·
· · ·· 7 7
· 7 7
⇒ (Z2)4 is a group of central type with coisotropic stabilizers
⇒ classification gives graph Γ′ quantum isomorphic to Γ
⇒ Γ′ coincides with a graph in [1] coming from the Mermin-Peres square
⇒ Pseudo-telepathy from the symmetries of classical magic squares
David Reutter Quantum graph isomorphisms March 20, 2018 22 / 23
[1] Atserias, Mancinska, Roberson, Samal, Severini and Varvitsiotis — Quantum [...] graph isomorphisms. 2017
![Page 155: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/155.jpg)
An example: binary magic squares (BMS)
BMS: a 3× 3 square with
{entries in {0, 1}
rows and columns add up to 0 mod 2Define a graph ΓBMS :
vertices: partial BMS — only one row or column filled — 24 vertices1 1 0· · ·· · ·
· · ·· · ·0 0 0
1 · ·0 · ·1 · ·
7 7
7 7
7 7
7 7
7 7
7 7
7 7
edge between two vertices if the partial BMS contradict each other
Bit-flip symmetries of this graph form a subgroup (Z2)4 ≤ Aut(ΓBMS).
generators:7 7 ·7 7 ·· · ·
· 7 7
· 7 7
· · ·
· · ·7 7 ·7 7 ·
· · ·· 7 7
· 7 7
⇒ (Z2)4 is a group of central type with coisotropic stabilizers
⇒ classification gives graph Γ′ quantum isomorphic to Γ
⇒ Γ′ coincides with a graph in [1] coming from the Mermin-Peres square
⇒ Pseudo-telepathy from the symmetries of classical magic squares
David Reutter Quantum graph isomorphisms March 20, 2018 22 / 23
[1] Atserias, Mancinska, Roberson, Samal, Severini and Varvitsiotis — Quantum [...] graph isomorphisms. 2017
![Page 156: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/156.jpg)
An example: binary magic squares (BMS)
BMS: a 3× 3 square with
{entries in {0, 1}
rows and columns add up to 0 mod 2Define a graph ΓBMS :
vertices: partial BMS — only one row or column filled — 24 vertices1 1 0· · ·· · ·
· · ·· · ·0 0 0
1 · ·0 · ·1 · ·
7 7
7 7
7 7
7 7
7 7
7 7
7 7
edge between two vertices if the partial BMS contradict each other
Bit-flip symmetries of this graph form a subgroup (Z2)4 ≤ Aut(ΓBMS).
generators:7 7 ·7 7 ·· · ·
· 7 7
· 7 7
· · ·
· · ·7 7 ·7 7 ·
· · ·· 7 7
· 7 7
⇒ (Z2)4 is a group of central type with coisotropic stabilizers
⇒ classification gives graph Γ′ quantum isomorphic to Γ
⇒ Γ′ coincides with a graph in [1] coming from the Mermin-Peres square
⇒ Pseudo-telepathy from the symmetries of classical magic squares
David Reutter Quantum graph isomorphisms March 20, 2018 22 / 23
[1] Atserias, Mancinska, Roberson, Samal, Severini and Varvitsiotis — Quantum [...] graph isomorphisms. 2017
![Page 157: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/157.jpg)
An example: binary magic squares (BMS)
BMS: a 3× 3 square with
{entries in {0, 1}
rows and columns add up to 0 mod 2Define a graph ΓBMS :
vertices: partial BMS — only one row or column filled — 24 vertices1 1 0· · ·· · ·
· · ·· · ·0 0 0
1 · ·0 · ·1 · ·
7 7
7 7
7 7
7 7
7 7
7 7
7 7
edge between two vertices if the partial BMS contradict each other
Bit-flip symmetries of this graph form a subgroup (Z2)4 ≤ Aut(ΓBMS).
generators:7 7 ·7 7 ·· · ·
· 7 7
· 7 7
· · ·
· · ·7 7 ·7 7 ·
· · ·· 7 7
· 7 7
⇒ (Z2)4 is a group of central type with coisotropic stabilizers
⇒ classification gives graph Γ′ quantum isomorphic to Γ
⇒ Γ′ coincides with a graph in [1] coming from the Mermin-Peres square
⇒ Pseudo-telepathy from the symmetries of classical magic squares
David Reutter Quantum graph isomorphisms March 20, 2018 22 / 23
[1] Atserias, Mancinska, Roberson, Samal, Severini and Varvitsiotis — Quantum [...] graph isomorphisms. 2017
![Page 158: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/158.jpg)
Summary
We have
described a framework for finite quantum set and graph theory
which links compact quantum group theory, fusion category theoryand quantum information
and applied it to classify quantum isomorphic graphs.
Many open questions:
Other physical applications?
Other theories based on finite quantum sets? Quantum orders?
Quantum combinatorics?
...
Thanks for listening!
David Reutter Quantum graph isomorphisms March 20, 2018 23 / 23
![Page 159: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/159.jpg)
Summary
We have
described a framework for finite quantum set and graph theory
which links compact quantum group theory, fusion category theoryand quantum information
and applied it to classify quantum isomorphic graphs.
Many open questions:
Other physical applications?
Other theories based on finite quantum sets? Quantum orders?
Quantum combinatorics?
...
Thanks for listening!
David Reutter Quantum graph isomorphisms March 20, 2018 23 / 23
![Page 160: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/160.jpg)
Summary
We have
described a framework for finite quantum set and graph theory
which links compact quantum group theory, fusion category theoryand quantum information
and applied it to classify quantum isomorphic graphs.
Many open questions:
Other physical applications?
Other theories based on finite quantum sets? Quantum orders?
Quantum combinatorics?
...
Thanks for listening!
David Reutter Quantum graph isomorphisms March 20, 2018 23 / 23
![Page 161: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/161.jpg)
Summary
We have
described a framework for finite quantum set and graph theory
which links compact quantum group theory, fusion category theoryand quantum information
and applied it to classify quantum isomorphic graphs.
Many open questions:
Other physical applications?
Other theories based on finite quantum sets? Quantum orders?
Quantum combinatorics?
...
Thanks for listening!
David Reutter Quantum graph isomorphisms March 20, 2018 23 / 23
![Page 162: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/162.jpg)
Summary
We have
described a framework for finite quantum set and graph theory
which links compact quantum group theory, fusion category theoryand quantum information
and applied it to classify quantum isomorphic graphs.
Many open questions:
Other physical applications?
Other theories based on finite quantum sets? Quantum orders?
Quantum combinatorics?
...
Thanks for listening!
David Reutter Quantum graph isomorphisms March 20, 2018 23 / 23
![Page 163: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/163.jpg)
Summary
We have
described a framework for finite quantum set and graph theory
which links compact quantum group theory, fusion category theoryand quantum information
and applied it to classify quantum isomorphic graphs.
Many open questions:
Other physical applications?
Other theories based on finite quantum sets? Quantum orders?
Quantum combinatorics?
...
Thanks for listening!
David Reutter Quantum graph isomorphisms March 20, 2018 23 / 23
![Page 164: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/164.jpg)
Summary
We have
described a framework for finite quantum set and graph theory
which links compact quantum group theory, fusion category theoryand quantum information
and applied it to classify quantum isomorphic graphs.
Many open questions:
Other physical applications?
Other theories based on finite quantum sets? Quantum orders?
Quantum combinatorics?
...
Thanks for listening!
David Reutter Quantum graph isomorphisms March 20, 2018 23 / 23
![Page 165: homepages.inf.ed.ac.ukhomepages.inf.ed.ac.uk/cheunen/cvqt/slides/Reutter.pdf · Overview This talk is based on joint work with Ben Musto and Dominic Verdon: A compositional approach](https://reader030.fdocuments.in/reader030/viewer/2022041013/5ec221f989961924c01c2ffa/html5/thumbnails/165.jpg)
Summary
We have
described a framework for finite quantum set and graph theory
which links compact quantum group theory, fusion category theoryand quantum information
and applied it to classify quantum isomorphic graphs.
Many open questions:
Other physical applications?
Other theories based on finite quantum sets? Quantum orders?
Quantum combinatorics?
...
Thanks for listening!
David Reutter Quantum graph isomorphisms March 20, 2018 23 / 23