Non-locality of
Symmetric States
Damian Markham
Joint work with:
Zizhu Wang
CNRS, LTCI
ENST (Telecom ParisTech), Paris
quant-ph/1112.3695
Motivation
• Non-locality as an interpretation/witness for
entanglement ‘types’?
(mutlipartite entanglment is a real mess)
• Different ‘types’ of non-locality?
(is non-locality also such a mess?)
Look at symmetric states – same tool
‘Majorana Representation’ used to study both….
Outline
1. Background (entanglement classes, non-locality,
Majorana representation for symmetric
states)
1. Hardy’s Paradox for symmetric states of n-qubits
2. Different Hardy tests for different entanglement
classes
Entanglement
Definition:
State is entangled iff NOT separapable
rSEP = å pis i1 Ä s i2 Ä… Ä s in
i
entangled
SEP
Entanglement
Types of entanglement
Dur, Vidal, Cirac, PRA 62, 062314 (2000)
In multipartite case, some states are incomparable, even under stochastic
Local Operations and Classical Communications (SLOCC)
1
SLOCC
2
Infinitely many different classes!
• Different resources for quantum information processing
• Different entanglement measures may apply for different types
Permutation Symmetric States
Symmetric under permutation of parties
  H n
Pij   
• 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
Permutation Symmetric States
Majorana representation
E. Majorana, Nuovo Cimento 9, 43 – 50 (1932)
j
1,2...n
=
eia
K
åh
1 1
Perm
h2 … hn
2
2
1
4
3
i  cos(i / 2) 0  ei sin(i / 2) 1
i
n
Permutation Symmetric States
Majorana representation
E. Majorana, Nuovo Cimento 9, 43 – 50 (1932)
j
1,2...n
=
eia
K
åh
Perm
1 1
h2 … hn
2
n
GHZ
2
1
000 + 111 )
(
=
2
= 2 2 å h1 h2 h3
Perm
3
1  1/ 2  0  1

2  1/ 2  0  ei 2 /3 1

2  1/ 2  0  ei 4 /3 1

Permutation Symmetric States
Majorana representation
E. Majorana, Nuovo Cimento 9, 43 – 50 (1932)
j
1,2...n
=
eia
K
åh
Perm
1 1
h2 … hn
2
n
nk
Dicke states
S(n,k) =
k
1
n
å
PERM
00...01..11
n-k
k
Permutation Symmetric States
Majorana representation
E. Majorana, Nuovo Cimento 9, 43 – 50 (1932)
j
1,2...n
=
=
e ia
K
e ia
K
åh
h2 … h n
1 1
Perm
2
åÄh
Perm
i
n
Ädi
Local unitary
i
U ÄU … ÄU j =
1
2
• Distribution of points alone
determines entanglement features
3
2
2
4
eia
K
åU h
1
U h2 … U hn
Perm
• Orthogonality relations
n
5
rotation of sphere
i    0
3
Äa
hi^ j
=0
a ³ n - di +1
¹0
a < n - di +1
Permutation Symmetric States
GHZ4
• Different degeneracy classes*
( di  )
T
S (4,1)  W4
Different entanglement ‘types’
(w.r.t. SLOCC)
• Different symmetries^
*T. Bastin, S. Krins, P. Mathonet, M. Godefroid, L. Lamata and E. Solano , PRL 103, 070503 (2009)
^D.M. PRA (2010)
Permutation Symmetric States
GHZ4
• Different degeneracy classes
( di  )
T
Different entanglement ‘types’
(w.r.t. SLOCC)
• Different symmetries
• NOTE: Almost identical states can be in different classes
Comparison to Spinor BEC
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)
Comparison to Spinor BEC
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)
Non-Locality
Measurement
a  0,1
basis
Measurement
outcome
V
V'
a0
a 1
H
A  0,1
b  0,1
H'
B  0,1
Non-Locality
Measurement
a  0,1
basis
Measurement
outcome
b  0,1
A  0,1
• Local Hidden Variable model
B  0,1
Non-Locality
Measurement
basis
m1  0,1
m2  0,1
M 1  0,1
mn  0,1
M 2  0,1
Measurement
outcome
• Local Hidden Variable model
M n  0,1
n Party case (Hardy’s Paradox)
• For all symmetric states
of n qubits (except Dicke states)
- set of probabilities which contradict LHV
- set of measurement which achieve these probabilities
measurement result
measurement basis
n Party case (Hardy’s Paradox)
• For all symmetric states
of n qubits (except Dicke states)
- set of probabilities which contradict LHV
- set of measurement which achieve these probabilities
measurement result
measurement basis
for some
,
n Party case (Hardy’s Paradox)
• For all symmetric states
of n qubits (except Dicke states)
- set of probabilities which contradict LHV
- set of measurement which achieve these probabilities
measurement result
measurement basis
for some
,
for same
,
n Party case (Hardy’s Paradox)
• For all symmetric states
of n qubits (except Dicke states)
- set of probabilities which contradict LHV
- set of measurement which achieve these probabilities
measurement result
measurement basis
for some
,
for same
,
CONTRADICTION!
n Party case (Hardy’s Paradox)
• For all symmetric states
of n qubits (except Dicke states)
- set of probabilities which contradict LHV
- set of measurement which achieve these probabilities
measurement result
measurement basis
n Party case (Hardy’s Paradox)
• Use Majorana representation
Y =
1
åh
1
K
Perm
= a h1
G
1
2...n
h 2 … hn
+ b h1^
1
Än
G
hi^ Y = 0
2...n
0 = h1
n Party case (Hardy’s Paradox)
• Use Majorana representation
Y =
å0
1
K Perm
=a 0 G
1
2...n
h2 … h n
+b 1 G
1
Än
2...n
hi^ Y = 0
G =
1
K
ån
1
Perm
1  1
0 = h1
… n n-1
n Party case (Hardy’s Paradox)
• Majorana representation always allows satisfaction of lower conditions, what
about the top condition?
- Must find, such that
n1 ¹ hi
?
1  1
0 = h1
n Party case (Hardy’s Paradox)
• Majorana representation always allows satisfaction of lower conditions, what
about the top condition?
- Must find, such that
n1 ¹ hi
- It works for all cases except product or Dicke
states
Dicke states too symmetric!
1  1
0 = h1
(almost)
All permutation symmetric states are non-local!
Testing Entanglement class
Testing Entanglement class
• Use the Majorana representation to add constraints implying degeneracy
Y =
=
1
K
1
K
å
h1 h2 … hn
Perm
åÄh
Perm
i
i
Ädi
n


i    0
i    0
i    0
  n  di  1
  n  di  1
Testing Entanglement class
• Use the Majorana representation to add constraints implying degeneracy
Y =
=
1
K
1
K
å
h1 h2 … hn
Perm
åÄh
Perm
i
i
Ädi
n


i    0
i    0
i    0
0 = hi
  n  di  1
  n  di  1
Testing Entanglement class
• Use the Majorana representation to add constraints implying degeneracy
Y =
=
1
K
1
K
å
h1 h2 … hn
Perm
åÄh
Perm
i
i
Ädi
n


i    0
i    0
i    0
  n  di  1
  n  di  1
0 = hi
Only satisfied by i
With degeneracy
di  d
Testing Entanglement class
• Use the Majorana representation to add constraints implying degeneracy
1
 
K
i k

Perm i 1
 di
i
i k
d
i 1
i
n
• Correlation persists to fewer sets, only for degenerate states.
0 = hi
Only satisfied by i
With degeneracy
di  d
e.g. Hardy Test for W class…
TW
Get REAL!!!
Get REAL!!!
• We really need to talk about inequalities
• ALL states with one MP of degeneracy
or greater violate for
• CanNOT be any longer true that ‘only’ states of the correct type violate
- states arbitrarily close which are in different class
Persistency of degenerate states
S (4,1)  W4
T
Conclusions
• Hardy tests for almost all symmetric states
• Different entanglement types
Different Hardy tests
- Non-locality to ‘witness’ / interpret entanglement types
- Different flavour of non-local arguments to Stabiliser
states/ GHZ states e.t.c.
• More?
- ‘Types’ of non-locality? (from operational perspective)
- Witness phase transitions by different locality tests?
- Meaning in terms of non-local games?
Thank you!
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