Damian Markham- Entanglement Symmetry of Permutation Symmetric States

8/3/2019 Damian Markham- Entanglement Symmetry of Permutation Symmetric States

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Entanglement Symmetry of

Permutation S mmetric States

Damian Markham

CNRS, LTCI ENST (Telecom ParisTech), Paris


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Motivation

What is the role of entanglement in many-body physics?


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Entanglement

Definition:

State is entangled iff NOT separapable

1 2 n

S E P i i i ip =

SEP

entangled


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Entanglement

Quantification

Monotone under Local Operations and Classical Communication

LOCC

ALICE BOB

ALICE BOB

Measure

Unitary

Phone

Measure

Unitary

Phone

ii

p,

:)(E

)(E iiiEp )(

{ }ii p,

Any function non-

increasing under LOCC

LOCC


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Entanglement

Quantification

Monotone under Local Operations and Classical Communication

Example:

minimum distance

to separable states

SEP

( ) : min ( | )SEP

E D

=( | )D


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Entanglement

Types of entanglement

Dur, Vidal, Cirac, PRA 62, 062314 (2000)

In multipartite case, some states are incomparable, even under stochasticLOCC (SLOCC)

Infinitely many different classes!

Different resources for quantum information processing

Different entanglement measures may apply for different types

1 2SLOCC


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Symmetry and Entanglement?

Why should we expect a relationship?

Symmetry breaking in critical phenomena related to long rangecorrelations entanglement is most quantum correlations

We see some ! - Change in entanglement properties at critical points in variouscases, e.g. In 1D Heisenberg chains

A. Osterloh et al Nature 416, 608 (2002)

L. Amico et al Rev. Mod. Phys. 80, 517-576 (2008)

Symmetry a useful tool for entanglement theory


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Symmetry and Entanglement?

Next...

Multiparty entanglement?

SLOCC classification play any role?

Deeper connection between symmetry and entanglement?

Look at symmetry under local unitaries of

permutation symmetric states


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Permutation Symmetric States

Symmetric under permutation of parties

nH

Occur as ground states e.g. of some Bose Hubbard models

Useful in a variety of Quantum Information Processing tasks

Experimentally accessible in variety of media

ij


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Majorana representation

E. Majorana, Nuovo Cimento 9, 43 50 (1932)

1 2

i

n

Perm

eK

=

cos( / 2) 0 sin( / 2) 1iii i ie

= +

12

3

4


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000 111+

1 2

i

n

Perm

eK

=

( )2 /32 1/ 2 0 1ie = +

1

2

3

1 2 3

2

2 2Perm

GHZ

=

=

( )1 1/ 2 0 1= +

( )4 /32 1/ 2 0 1ie = +


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1 2

i

n

Perm

eK

= n k

1( , ) 00...01..11

PERM n k k

S n kn

=

Dicke states

k


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Local Unitary (LU) Rotation of sphere

1 2

i

n

Perm

eU U U U U U

K

=

Symmetries under LU are symmetries of points under SO(3)

Visualise the geometric measure of entanglement

LU symmetries imply equivalence of different entanglement measures

Different symmetries (LU) different entanglement types (w.r.t.

SLOCC)


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Geometric Measure of Entanglement


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Distance (overlap) to closest product state

( | )D

22 |||

1logmin:)(|

=

prodGE

SEP


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)(|

1log:)(| 2

=GE

2|||max: =

rod

( | )D

SEP


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)(|

1log:)(| 2

=GE

2|||max: =

rod

( | )D

For

only need to look for *

SEP

..=

* R. Hubner, M. Kleinmann, T.-C. Wei, C. Gonzlez-Guilln, O. Guhne, PRA 80 032324 (2008)

1 2

i

n

Perm

e

K

=


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)(|

1log:)(| 2

=GE

2|||max: =

rod

( | )D

For

only need to look for *

SEP

..=

* R. Hubner, M. Kleinmann, T.-C. Wei, C. Gonzlez-Guilln, O. Guhne, PRA 80 032324 (2008)

1 2

i

n

Perm

e

K

=

2 2 2 2

1 21 ! max | | | | | | ... | | | sym nn

K =


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Majorana Representation

Closest product state is one point

..=

Calculating ->Maximising product of angles

- Symmetry of points helps calculate entanglement

- Most entangled most spread *

)(|

1log:)(| 2

=GE

2|||max: =

prod

2 2 2 2

1 2

1! max | | | | | | ... | | |

sym n

n

=

(| )GE

*M. Aulbach, DM and M. Murao, in preperation


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LU symmetry and equivalence of different

measures


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measures

Hierarchy of entanglement measures*

Geometric Measure

22|||

1logmin:)(|

prodGE

=

)()()( GR EEr

Relative entropy of entanglement

Logarithmic Robustness

))2log2(log(:)||(

)||(min:)(

=

=

trS

sepR SE

septt

t

++

=

+=

)()1(

1s.t.

)minlog(1:)r(

SEP

( | )D

*M. Hayashi, D. Markham, M. Murao , M. Owari and S. Virmani PRL 96 (2006) 040501


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measuresWho cares if these measures are equal?!?

Different operational meanings

e.g. Bounds channel capacitiesGeometric Measure

22 |||

1logmin:)(|

prodGE

=

information theoretic interpretations

robustness of entanglement againstnoise

Strategies for LOCC discrimination (access of information) *

Constuction of optimal entanglement witnesses **M. Hayashi, D. Markham, M. Murao , M. Owari and S. Virmani PRA 77 (2008) 012104



))2log2(log(:)||(

)||(min:)(

=

=

trS

sepR SE

septt

t

++

=

+=

)()1(

1s.t.

)minlog(1:)r(


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measures

Hierarchy of entanglement measures

Geometric Measure

22 |||

1logmin:)(|

prodGE

=

)()()( GR EEr



))2log2(log(:)||(

)||(min:)(

=

=

trS

sepR SE

septt

t

++

=

+=

)()1(

1s.t.

)minlog(1:)r(

SEP

( | )D


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measures


Geometric Measure

22 |||

1logmin:)(|

prodGE

=

( | ') ( ) ( ) ( )r R G D r E E



))2log2(log(:)||(

)||(min:)(

=

=

trS

sepR SE

septt

t

++

=

+=

)()1(

1s.t.

)minlog(1:)r(

SEP

( | )D

'

( | ')D

1( )

(1 )' t sep

t +

+=

( | ') log(1 )rD t = +


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measures


Geometric Measure

22 |||

1logmin:)(|

prodGE

=

( | ') ( ) ( ) ( )r R G D r E E



))2log2(log(:)||(

)||(min:)(

=

=

trS

sepR SE

septt

t

++

=

+=

)()1(

1s.t.

)minlog(1:)r(

SEP

( | )D

'

( | ')D

1( )

(1 )' t sep

t +

+=

( | ') log(1 )

( )

r

G

D t

E

= +

=


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measures

Twirl over a local group on the solution to the geometric measure

Geometric Measure

2 2

1(| ) : min log

| | |G

prodE

=

+= dUUU'

=i

ii PP

where and isprojection onto invariant subspace (ShuruuuU = ... iP


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measures

Twirl over a local group on the solution to the geometric measure

Geometric Measure

2 2

1(| ) : min log

| | |G

prodE

=

+= dUUU'

=i

ii PP

where and isprojection onto invariant subspace (Shur

Thus if

uuuU = ... iP

=1P

+= 22 1'

( += )()( 212' GG EE1 ( )

(1 )' t sept + +=( | ') log(1 ) ( )r G D t E = + =

M. Hayashi, D. Markham, M. Murao , M. Owari and S. Virmani PRA 77 (2008) 012104


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measures

Majorana Representation

If a permutation symmetric state has Majorana Representation whichis invariant under some subgroup , then this state satisfies

(3) X SO

Can systematically check all subgroups of SO(3) to find all such states

O(2), SO(2), Cn, Dn, T, O, Y

( ) ( ) ( )R Gr E E = =


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measures


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LU Symmetry and Entanglement types


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For all nStates of different symmetries

SO(2),Dn, T, O, Y

are SLOCC inconvertible!i.e. Different types of entanglement

Geometric properties of Majorana points underlie SLOCC classes


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E.g. n=4

Multiplicity of points cannot change under SLOCC

^T. Bastin, S. Krins, P. Mathonet, M. Godefroid, L. Lamata and E. Solano , PRL 103, 070503 (2009)


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E.g. n=4


^T. Bastin, S. Krins, P. Mathonet, M. Godefroid, L. Lamata and E. Solano , PRL 103, 070503 (2009)


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E.g. n=4


Minumum size of product state decomposition (Shmidt number r) cannot

change*

^T. Bastin, S. Krins, P. Mathonet, M. Godefroid, L. Lamata and E. Solano , PRL 103, 070503 (2009)*Dur, Vidal, Cirac, PRA 62, 062314 (2000)

1 2

1

rn

i i i

i

=

= log( )GE r( ) 3r T ( ) 2r GHZ =


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Summary

Local Unitary (LU) Rotation of sphere

1 2

i

n

Perm

eU U U U U U

K

=

Symmetries under LU are symmetries of points under SO(3)

Visualise the geometric measure of entanglement

LU symmetries imply equivalence of different entanglement measures

Different symmetries (LU) different entanglement types (w.r.t.

SLOCC)


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Comparison to Spinor BEC

n spin

Geometric measure of

entanglement

single spin s=n/2

Distance from spin coherent

state

LU symmetry and equality of

entanglement measures

Different symmetries, differententanglement types

Exact condition for inert states

Different symmetries, differentphases


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E.g. S=2

R. Barnett, A. Turner and E. Demler, PRL 97, 180412 (2007)

Phase diagram for spin 2 BEC in single optical trap

(Fig taken from PRL 97, 180412)


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E.g. S=2





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E.g. S=2


CAUTION: notentanglement if single system




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E.g. S=2


CAUTION: notentanglement if single system ... unless it is

(if made up from spin )




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Conclusions

Relation between LU symmetry and entanglement

- Geometric measure of entanglement- Equivalence of measures

- LU symmetries vs. types of entanglement

Intriguing mirrors in spinor BEC

MORE?- Generalisation?

- Entanglement as order parameter for symmetry

breaking?- Entanglement phase transitions?

Damian Markham- Entanglement Symmetry of Permutation Symmetric States

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