On Minimum Reversible Entanglement Generating Sets
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On Minimum Reversible Entanglement Generating Sets
Fernando G.S.L. Brandão
Cambridge 16/11/2009
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Quantum Entanglement
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Quantum Entanglement
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Quantum Entanglement
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Quantum Entanglement
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Quantum Entanglement
Cannot be created by local operations and classical communication (LOCC)
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A B
LOCC asymptotic entanglement transformations
nknLOCC
n
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LOCC asymptotic entanglement transformations
A B
nD n
n
nn
LOCCn
LOCC
n
n
lim,0)),((minlim
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Bipartite pure state entanglement transformations
• Transformations are reversible (Bennett, Bernstein, Popescu, Schumacher 96)
• Unique entanglement measure
Entropy of Entanglement:
• The asymptotic limit is crucial!
)()( ASE
)()(
)()(
E
LOCCE
ELOCC
E
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Bipartite pure state entanglement transformations
• Transformations are reversible (Bennett, Bernstein, Popescu, Schumacher 96)
• Unique entanglement measure
Entropy of Entanglement:
• The asymptotic limit is crucial!
)()( ASE
)()(
)()(
E
LOCCE
ELOCC
E
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Bipartite pure state entanglement transformations
• Transformations are reversible (Bennett, Bernstein, Popescu, Schumacher 96)
• Unique entanglement measure
Entropy of Entanglement:
• The asymptotic limit is crucial
)()( ASE
)()(
)()(
E
LOCCE
ELOCC
E
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Multipartite pure state entanglement transformations
Alice
Bob
Charlie
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Multipartite pure state entanglement transformations
• There are inequivalent types of entanglement
ABCBACCABEPREPREPR 000
Alice
Bob
Charlie
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Multipartite pure state entanglement transformations
• ???BCACAB E
BC
E
AC
E
ABLOCCABCEPREPREPR
Alice
Bob
Charlie
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Multipartite pure state entanglement transformations
•
BCACAB E
BC
E
AC
E
ABLOCCABCEPREPREPRGHZ
Alice
Bob
Charlie
(Linden, Popescu, Schumacher, Westmoreland 99)
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•
• ??
Multipartite pure state entanglement transformations
Alice
Bob
Charlie
ABCBCACAB E
ABC
E
BC
E
AC
E
ABLOCCABCGHZEPREPREPR
(Linden, Popescu, Schumacher, Westmoreland 99)
BCACAB E
BC
E
AC
E
ABLOCCABCEPREPREPRGHZ
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Multipartite pure state entanglement transformations
•
•
ABABAB r
AC
r
AC
r
ABLOCCABCEPREPREPRGHZ
Alice
Bob
Charlie
(Linden, Popescu, Schumacher, Westmoreland 99)
(Acin, Vidal, Cirac 02)
ABCBCACAB E
ABC
E
BC
E
AC
E
ABLOCCABCGHZEPREPREPR
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(Bennett, Popescu, Rohrlich, Smolin, Thapliyal 99)
NkABCk 1
• Is there a finite set of states such that
• This is the MREGS question...
Bob
Alice Charlie
Multipartite pure state entanglement transformations
kr
ABCk
N
kLOCCABC
1
???THIS TALK GOAL: To expose poor state of MREGS (Minimum Reversible Entanglement Generating Set)
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MREGS
• We say is a MREGS if
and N is the minimum number for which this holds
• Conjecture: There is no finite MREGS
• Rest of the talk: two arguments supporting the conjecture
NkABCk 1
kr
ABCk
N
kLOCCABC
1
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No MREGS under 1-LOCC
Alice
Bob
Charlie
• Theo 1: Under 1-way LOCC (Alice to Bob and Bob to Charlie), there is no finite MREGS for tripartite states of qubits (2 x 2 x 2 states)
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Sometimes1-LOCC is enough
• For bipartite pure states 1-LOCC is enough (Bennett, Bernstein, Popescu, Schumacher 96)
• For the only two known classes of multipartite states for which a MREGS exist, 1-LOCC is enough
1. Schmidt decomposable states(Bennett, Popescu, Rohrlich, Smolin, Thapliyal):
2. The family (Vidal, Dür, Cirac):
k
kkkk cba ,,
22111000 10 cc
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Sometimes1-LOCC is enough
• For bipartite pure states 1-LOCC is enough (Bennett, Bernstein, Popescu, Schumacher 96)
• For the only two known classes of multipartite states for which a MREGS exist, 1-LOCC is enough
1. Schmidt decomposable states(Bennett, Popescu, Rohrlich, Smolin, Thapliyal):
2. The family (Vidal, Dür, Cirac):
k
kkkk cba ,,
22111000 10 cc
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Sometimes1-LOCC is enough
• For bipartite pure states 1-LOCC is enough (Bennett, Bernstein, Popescu, Schumacher 96)
• For the only two known classes of multipartite states for which a MREGS exist, 1-LOCC is enough
1. Schmidt decomposable states(Bennett, Popescu, Rohrlich, Smolin, Thapliyal):
2. The family (Vidal, Dür, Cirac 01):
k
kkkk cba ,,
22111000 10 cc
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• (proof theo 1) Suppose there is a finite MREGS:
kr
ABCk
N
kLOCCABC
11
From tripartite pure to bipartite mixed states
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• (proof theo 1) Suppose there is a finite MREGS:
• Since Charlie cannot communicate, we must have
kr
ABCk
N
kLOCCABC
11
krABk
N
kLOCCAB
,
11
From tripartite pure to bipartite mixed states
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From tripartite pure to bipartite mixed states
• (proof theo 1) Suppose there is a finite MREGS:
• Since Charlie cannot communicate, we must have
• Thus there must exist a mixed bipartite MREGS under 1-LOCC... but we can show it cannot exist!
kr
ABCk
N
kLOCCABC
11
krABk
N
kLOCCAB
,
11
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k-Extendible States
• We say a state is k-extendible if there is a
state such that for all j
• Only separable states are k-extendable for every k (Raggio and Werner 89)
• ...and for every k there are k-extendable yet entangled states
AB
kBAB ...1
ABABj
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k-Extendible States
• We say a state is k-extendable if there is a
state such that for all j
• Only separable states are k-extendible for every k (Raggio and Werner 89)
• ...and for every k there are k-extendable yet entangled states
AB
kBAB ...1
ABABj
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k-Extendible States
• We say a state is k-extendable if there is a
state such that for all j
• Only separable states are k-extendable for every k (Raggio and Werner 89)
• ...and for every k there are k-extendible yet entangled states
AB
kBAB ...1
ABABj
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k-Extendability is preserved by 1-LOCC
• ( ) the set of k-extendible states is preserved under 1-LOCC and under tensoring
• Let kmax be the max. k such that all are k-ext.
• Take to be a (kmax +1)-ext. state. QED
1-ext2-extSEP
ABk ,
AB
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k-Extendability is preserved by 1-LOCC
• ( ) the set of k-extendible states is preserved under 1-LOCC and under tensoring
• Let kmax be the max. k such that all are k-ext.
• Take to be a (kmax +1)-ext. state. QED
1-ext2-extSEP
ABk ,
AB
krABk
N
kLOCC
AB
,1
1
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Intermezzo: Interconverting two useless channels
• We just saw there are infinitely many different types of 1-LOCC undistillable states (namely, k-extendible states)
• By Jamiolkowski/Choi isomorphism, we find there are infinitely many inequivalent classes of zero-capacity channels too: There is an infinite
sequence such that for all n,
• E.g. Erasure channel with 1/n erasure probability.
NkBAk HH :
knk 1
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Intermezzo: Interconverting two useless channels
• We just saw there are infinitely many different types of 1-LOCC undistillable states (namely, k-extendible states)
• By Jamiolkowski/Choi isomorphism, we find there are infinitely many inequivalent classes of zero-capacity channels too: There is an infinite
sequence such that for all n,
• E.g. Erasure channel with 1/n erasure probability.
NkBAk HH :
knk 1
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No MREGS under LOCC?
Alice
Bob
Charlie
In the general case of unrestricted LOCC communication among the parties, all proof fails completely Reason: general LOCC do not preserve the sets of k-extendible states.
Is there a good replacement for k-ext. states??
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No MREGS under LOCC?
Alice
Bob
Charlie
• Theo 2: Assuming a certain conjecture about bipartite mixed states, there is no MREGS for tripartite systems (already for 3 x 3 x 2).
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There are infinitely many inequivalent types of bipartite entanglement, i.e. such that for all k, n
1kk
1 kn
k
The Conjecture
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• We know two classes: 1. distillable states:
2. bound entanglement: (Horodecki, Horodecki, Horodecki 98)
EPREPRn
EPREPRn
Bound Entanglement
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• We know two classes: 1. distillable states:
2. bound entanglement: (Horodecki, Horodecki, Horodecki 98)
• The conjecture is really about bound entanglement: We want a sequence of bound entangled states s.t.
EPREPRn
EPREPRn
1kk
1 kn
k
Bound Entanglement
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• Why is it reasonable?
“The same evidences that we have for bound entangled Werner states with a non-positive partial transpose apply to the conjecture.”
• Namely
Let For any n there is a non-zero interval (a, b) for which
But the interval might shrink to zero when n grows....
0),,( bappn
p
)1( ppp
Supporting the Conjecture
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• (proof sketch) Assume there is a finite MREGS
• As before, the strategy is to relate it to a problem about bipartite mixed states...
• The basic idea dates back to Linden, Popescu, Schumacher, Westmoreland 99 who used it to show that the GHZ state is not equivalent to EPR pairs.
kr
ABCk
N
kLOCCABC
1
From Tripartite Pure to Bipartite Mixed, Again
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The Relative Entropy Must Be Preserved
• Linden, Popescu, Schumacher, Westmoreland proved that
implies
where the regularized relative entropy of entanglement reads
krABkkRABR EE ,)(
)||(min1lim:)(
nABSnABR S
nE
kr
ABCk
N
kLOCCABC
1
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Some New Entanglement Measures
• We do the same, but for infinitely many
related measures. Let s.t. and define
Where the asymptotic orbit of reads
SLOCCtrO nnk )),((/)(::)(
)||(min1lim:)()(
nABOnABk S
nE
k
1kk 1
kn
k
k
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The Relative Entropies Must Be Preserved
• We can show: If
For all k
The proof is an easy adaptation of the result of Popescu et al. (the measures have all the nice properties: monotone under LOCC, non-lockable, asymptotically continuous, subadditive, ...)
jr
ABjjkABk EE ,)(
kr
ABCk
N
kLOCCABC
1
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One Last Nice Property
• From a result of B., Plenio 09 on extensions of quantum Stein’s Lemma we find
• The proof is the same as for the regularized relative entropy of entanglement. See also M. Piani 09.
0)()( 1 kk EO
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The final part
• Assuming MREGS, we have
1)
2)
• Therefore there for every k there is a j such that
• But this cannot be, as by assumption there are only finitely many j. QED
)()(0 ,, ABjkjrABjjkkk ErEE j
0)(,0)( ,1,
ABjkABjk EE
j ABjkjrABjjkkk ErEE j )()(0 ,1,11
j
k
r
ABCj
N
jLOCCABC
1
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The final part
• Assuming MREGS, we have
1)
2)
• Therefore there for every k there is a j such that
• But this cannot be, as by assumption there are only finitely many j. QED
)()(0 ,, ABjkjrABjjkkk ErEE j
0)(,0)( ,1,
ABjkABjk EE
j ABjkjrABjjkkk ErEE j )()(0 ,1,11
j
k
r
ABCj
N
jLOCCABC
1
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The final part
• Assuming MREGS, we have
1)
2)
• Therefore for every k there is a j such that
• But this cannot be, as by assumption there are only finitely many j. QED
)()(0 ,, ABjkjrABjjkkk ErEE j
0)(,0)( ,1,
ABjkABjk EE
j ABjkjrABjjkkk ErEE j )()(0 ,1,11
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The final part
• Assuming MREGS, we have
1)
2)
• Therefore for every k there is a j such that
• But this cannot be, as by assumption there are only finitely many j. QED
)()(0 ,, ABjkjrABjjkkk ErEE j
0)(,0)( ,1,
ABjkABjk EE
j ABjkjrABjjkkk ErEE j )()(0 ,1,11
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Conclusion and Open Questions
• We showed that there is no finite MREGS
1. Under 1-LOCC 2. Assuming there are infinitely many classes of bipartite bound entanglement
• Open questions:
1. Is backward classical communication helpful?
2. Can we prove there are more than 2 types of of bound entanglement? Related to NPPTBE!
3. Is there a finite MREGS?
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Thank you!