A diagrammatic axiomatisation of the GHZ and W quantum states
Transcript of A diagrammatic axiomatisation of the GHZ and W quantum states
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A diagrammatic axiomatisationof the GHZ and W quantum states
Amar Hadzihasanovic
University of Oxford
Oxford, 17 July 2015
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The unhelpful third party
GHZ: |000〉+ |111〉
W: |001〉+ |010〉+ |100〉
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The unhelpful third party
GHZ: |000〉+ |111〉
W: |001〉+ |010〉+ |100〉
![Page 4: A diagrammatic axiomatisation of the GHZ and W quantum states](https://reader031.fdocuments.in/reader031/viewer/2022040322/62481d2a4b2854331129bde8/html5/thumbnails/4.jpg)
The unhelpful third party
GHZ: |000〉+ |111〉
W: |001〉+ |010〉+ |100〉
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Only the arity counts
By map-state duality, a tripartite state is the same as a binaryoperation
7→ ←[000 + 111|0〉〈00|+|1〉〈11| |000〉+|111〉
Bob & Aleks, 2010: we can associate commutative Frobeniusalgebras (with different properties) to the GHZ and W states
=
GHZ and W as building blocks for higher SLOCC classes?
![Page 6: A diagrammatic axiomatisation of the GHZ and W quantum states](https://reader031.fdocuments.in/reader031/viewer/2022040322/62481d2a4b2854331129bde8/html5/thumbnails/6.jpg)
Only the arity counts
By map-state duality, a tripartite state is the same as a binaryoperation
7→ ←[000 + 111|0〉〈00|+|1〉〈11| |000〉+|111〉
Bob & Aleks, 2010: we can associate commutative Frobeniusalgebras (with different properties) to the GHZ and W states
=
GHZ and W as building blocks for higher SLOCC classes?
![Page 7: A diagrammatic axiomatisation of the GHZ and W quantum states](https://reader031.fdocuments.in/reader031/viewer/2022040322/62481d2a4b2854331129bde8/html5/thumbnails/7.jpg)
Only the arity counts
By map-state duality, a tripartite state is the same as a binaryoperation
7→ ←[000 + 111|0〉〈00|+|1〉〈11| |000〉+|111〉
Bob & Aleks, 2010: we can associate commutative Frobeniusalgebras (with different properties) to the GHZ and W states
=
GHZ and W as building blocks for higher SLOCC classes?
![Page 8: A diagrammatic axiomatisation of the GHZ and W quantum states](https://reader031.fdocuments.in/reader031/viewer/2022040322/62481d2a4b2854331129bde8/html5/thumbnails/8.jpg)
Axiomatise
Goal: An as-complete-as-possible diagrammatic axiomatisation ofthe relations between GHZ and W
Desiderata (the basic ZX calculus meets these!):
a faithful graphical representation of symmetries (if somethinglooks symmetrical, it better be)
the axioms should look familiar to algebraists and/ortopologists
![Page 9: A diagrammatic axiomatisation of the GHZ and W quantum states](https://reader031.fdocuments.in/reader031/viewer/2022040322/62481d2a4b2854331129bde8/html5/thumbnails/9.jpg)
Axiomatise
Goal: An as-complete-as-possible diagrammatic axiomatisation ofthe relations between GHZ and W
Desiderata (the basic ZX calculus meets these!):
a faithful graphical representation of symmetries (if somethinglooks symmetrical, it better be)
the axioms should look familiar to algebraists and/ortopologists
![Page 10: A diagrammatic axiomatisation of the GHZ and W quantum states](https://reader031.fdocuments.in/reader031/viewer/2022040322/62481d2a4b2854331129bde8/html5/thumbnails/10.jpg)
Axiomatise
Goal: An as-complete-as-possible diagrammatic axiomatisation ofthe relations between GHZ and W
Desiderata (the basic ZX calculus meets these!):
a faithful graphical representation of symmetries (if somethinglooks symmetrical, it better be)
the axioms should look familiar to algebraists and/ortopologists
![Page 11: A diagrammatic axiomatisation of the GHZ and W quantum states](https://reader031.fdocuments.in/reader031/viewer/2022040322/62481d2a4b2854331129bde8/html5/thumbnails/11.jpg)
The ZW calculus
Result: the ZW calculus is complete for the category of abeliangroups generated by Z⊕ Z through tensoring†
† “qubits with integer coefficients”, embedding into finite-dimcomplex Hilbert spaces through the inclusion Z ↪→ C
Warning
I’ll show you a different (but equivalent) version from the one inthe paper
![Page 12: A diagrammatic axiomatisation of the GHZ and W quantum states](https://reader031.fdocuments.in/reader031/viewer/2022040322/62481d2a4b2854331129bde8/html5/thumbnails/12.jpg)
The ZW calculus
Result: the ZW calculus is complete for the category of abeliangroups generated by Z⊕ Z through tensoring†
† “qubits with integer coefficients”, embedding into finite-dimcomplex Hilbert spaces through the inclusion Z ↪→ C
Warning
I’ll show you a different (but equivalent) version from the one inthe paper
![Page 13: A diagrammatic axiomatisation of the GHZ and W quantum states](https://reader031.fdocuments.in/reader031/viewer/2022040322/62481d2a4b2854331129bde8/html5/thumbnails/13.jpg)
The ZW calculus
Result: the ZW calculus is complete for the category of abeliangroups generated by Z⊕ Z through tensoring†
† “qubits with integer coefficients”, embedding into finite-dimcomplex Hilbert spaces through the inclusion Z ↪→ C
Warning
I’ll show you a different (but equivalent) version from the one inthe paper
![Page 14: A diagrammatic axiomatisation of the GHZ and W quantum states](https://reader031.fdocuments.in/reader031/viewer/2022040322/62481d2a4b2854331129bde8/html5/thumbnails/14.jpg)
The construction of ZW
1 Layer one: Cross
2 Layer two: Even
3 Layer three: Odd
4 Layer four: Copy
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A matter of space
The new generators: cup, cap, symmetric braiding, crossing
=
What they satisfy:Cup + cap + braiding: zigzag equations + symmetricReidemeister I, II, III
== ,=
= , =
,
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A matter of space
The new generators: cup, cap, symmetric braiding, crossing
=
What they satisfy:Cup + cap + braiding: zigzag equations + symmetricReidemeister I, II, III
== ,=
= , =
,
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Who framed Reidemeister?
Cup + cap + crossing: symmetric Reidemeister II, III;Reidemeister I to be replaced by
=
(logic of blackboard-framed links, but with a symmetric braiding)
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Who framed Reidemeister?
Cup + cap + crossing: symmetric Reidemeister II, III;Reidemeister I to be replaced by
=
(logic of blackboard-framed links, but with a symmetric braiding)
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The construction of ZW
1 Layer one: Cross
2 Layer two: Even
3 Layer three: Odd
4 Layer four: Copy
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Black dots
The new generator: W algebra
What it satisfies:
= =,
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Black dots
The new generator: W algebra
What it satisfies:
= =,
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The W bialgebra...
What it satisfies (continued):
= =,
= = =, ,
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The W bialgebra...
What it satisfies (continued):
= =,
= = =, ,
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...well - Hopf algebra
What it satisfies (finally):
=
Will be provable:
=
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...well - Hopf algebra
What it satisfies (finally):
=
Will be provable:
=
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From ZW to ZX
One can build a gate
:=
This is actually the ternary red gate of the ZX calculus, aka Z2
on the computational basis
(SLOCC-equivalent to GHZ)
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From ZW to ZX
One can build a gate
:=
This is actually the ternary red gate of the ZX calculus, aka Z2
on the computational basis
(SLOCC-equivalent to GHZ)
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Fun fact
Then, interpreted in VecR,
p q
represents multiplication in Clp,q(R), the real Clifford algebra withsignature (p, q)
braiding : crossing = commutation : anticommutation
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Fun fact
Then, interpreted in VecR,
p q
represents multiplication in Clp,q(R), the real Clifford algebra withsignature (p, q)
braiding : crossing = commutation : anticommutation
![Page 30: A diagrammatic axiomatisation of the GHZ and W quantum states](https://reader031.fdocuments.in/reader031/viewer/2022040322/62481d2a4b2854331129bde8/html5/thumbnails/30.jpg)
The construction of ZW
1 Layer one: Cross
2 Layer two: Even
3 Layer three: Odd
4 Layer four: Copy
![Page 31: A diagrammatic axiomatisation of the GHZ and W quantum states](https://reader031.fdocuments.in/reader031/viewer/2022040322/62481d2a4b2854331129bde8/html5/thumbnails/31.jpg)
The X gate
The new generator: Pauli X
What it satisfies:
== ,
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The X gate
The new generator: Pauli X
What it satisfies:
== ,
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Purity
So far: only purely even/purely odd maps works for fermions
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The construction of ZW
1 Layer one: Cross
2 Layer two: Even
3 Layer three: Odd
4 Layer four: Copy
![Page 35: A diagrammatic axiomatisation of the GHZ and W quantum states](https://reader031.fdocuments.in/reader031/viewer/2022040322/62481d2a4b2854331129bde8/html5/thumbnails/35.jpg)
White dots
The new generator: GHZ algebra
What it satisfies:
= =,
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White dots
The new generator: GHZ algebra
What it satisfies:
= =,
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If it’s black, copy it
What it satisfies (continued):
= =, ,
= ,=
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If it’s black, copy it
What it satisfies (continued):
= =, ,
= ,=
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Detach
What it satisfies (finally):
=
(crossing elimination rule)
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What next?
1 Make it more topological.So far, quite satisfactory understanding up to layer two.
This might help us
2 Find better normal forms.The one used in the proof is as informative as vector notation.
Everything disconnectable should be disconnected!
3 Understand how expressive each layer is.Layer two already contains both 3-qubit SLOCC classes.
![Page 41: A diagrammatic axiomatisation of the GHZ and W quantum states](https://reader031.fdocuments.in/reader031/viewer/2022040322/62481d2a4b2854331129bde8/html5/thumbnails/41.jpg)
What next?
1 Make it more topological.So far, quite satisfactory understanding up to layer two.
This might help us
2 Find better normal forms.The one used in the proof is as informative as vector notation.
Everything disconnectable should be disconnected!
3 Understand how expressive each layer is.Layer two already contains both 3-qubit SLOCC classes.
![Page 42: A diagrammatic axiomatisation of the GHZ and W quantum states](https://reader031.fdocuments.in/reader031/viewer/2022040322/62481d2a4b2854331129bde8/html5/thumbnails/42.jpg)
What next?
1 Make it more topological.So far, quite satisfactory understanding up to layer two.
This might help us
2 Find better normal forms.The one used in the proof is as informative as vector notation.
Everything disconnectable should be disconnected!
3 Understand how expressive each layer is.Layer two already contains both 3-qubit SLOCC classes.
![Page 43: A diagrammatic axiomatisation of the GHZ and W quantum states](https://reader031.fdocuments.in/reader031/viewer/2022040322/62481d2a4b2854331129bde8/html5/thumbnails/43.jpg)
Extensions
1 From integers to real numbers.Signed metric on wires?
|0〉 〈0|+ eλ |1〉 〈1|λ
2 Complex phases.Topology might again give some suggestions!
=π phase asπ
2?=
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Extensions
1 From integers to real numbers.Signed metric on wires?
|0〉 〈0|+ eλ |1〉 〈1|λ
2 Complex phases.Topology might again give some suggestions!
=π phase asπ
2?=
![Page 45: A diagrammatic axiomatisation of the GHZ and W quantum states](https://reader031.fdocuments.in/reader031/viewer/2022040322/62481d2a4b2854331129bde8/html5/thumbnails/45.jpg)
Thank you for your attention!
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p1 pq
n
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b1,1bq,n−1
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q∑i=1
(−1)pi mi |bi ,1 . . . bi ,n〉