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Relative entropy and squashed entanglement
Ke Li (CQT, NUS) Andreas Winter (Universitat Autònoma de Barcelona)
QIP2013 Beijing
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Entanglement and entanglement measures
, ,c d dE E K
Entanglement is an important concept and resource. It isthe most outstanding non-classical feature of compound states that can’t be expressed as mixture of product states.
Entanglement measures: To understand entanglement, we need entanglement measures with good properties.
- operational ones: …- abstract ones: …, , ,f r sq nE E E E
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Squashed entanglement and
relative entropy of entanglement
Squashed entanglement (Tucci ’99, ’02; Christandl & Winter ’04)
- It has many nice properties (monogamy, additivity, etc) and operational meaning (Koashi & Winter ’04; Christandl & Winter ’04 ; Devetak & Yard ’08)
Relative entropy of entanglement (Vedral et al ’97, ’98)
with
- Regularized version admits operational meaning. (Brandao & Plenio ’08, ’10)
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Post-measurement relative entropy of entanglement, with respect to restricted measurement classes (Piani ’09)
Here M is a class of measurement, such as
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Outline
Results:
1. Monogamy relation for relative entropy of entanglement
2. Commensurate lower bound for squashed entanglement
3. Properties of : asymptotic continuity and evaluation on maximally entangled states and pure states.
4. Comparisons between entanglement measures.
Proofs (of 1 and 2)
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Monogamy relation for relative entropy of entanglement
For an entanglement measure f, one would expect the monogamy
This is true for , but fails for (along with most other EMs).Counterexample: anti-symmetric states! (Christandl, Schuch, Winter ’10)
Properly weakened monogamy relation:
Theorem 1 For every tripartite quantum state , we have
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Commensurate lower bound for squashed entanglement
Squashed entanglement is faithful. (Brandão, Christandl, Yard ’10)
The main result of the proof is the following 1-LOCC trace-normbound:
where
ABρ( ) 0sq ABE ρ > ⇔
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Commensurate lower bound for squashed entanglement
We provide a 1-LOCC relative entropy lower bound for
Theorem 2 For every quantum state , we have
Strong subadditivity: (Lieb, Ruskai ’73)
Corollary (Refinement of strong subadditivity):
(note: )
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Commensurate lower bound for squashed entanglement
Recovering the 1-LOCC trace-norm bound:applying Pinsker’s inequality ,
we are able to recover the 1-LOCC trace-norm bound (with slightly better constant factor) :
It is asymptotically normalized: for maximally entangled state and pure state ,
About the new bound:
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Properties of
Asymptotic continuity
Proposition 3 Let be two states of dimension , with Then
Evaluation on maximally entangled states and pure states
Proposition 4 For rank-d maximally entangled state and pure state ,
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Comparisons between entanglement measures
We are mainly interested in two families of entanglement measures:
Squashed-like measures { }
E r families { }
- Conditional entanglement of mutual information (Yang, Horodecki, Wang ’08)
- C-squashed entanglement (Yang at el ’07)
- Relatives of relative entropy of entanglement (Piani ’09)
,
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Comparisons between entanglement measures
(focusing mainly on regularized versions)
We obtain:
Previously known ones:
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Proofs of Theorem 1 and Theorem 2
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Quantum hypothesis testing with one-way LOCC operations
Consider the setting of hypothesis testingtwo hypotheses: or tests allowed: on ; 1-LOCC implementable
two errors: ,
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Quantum hypothesis testing with one-way LOCC operations
For any 1-LOCC , there exist 1-LOCC tests such that
Quantum Stein’s Lemma(Hiai, Petz ’91)
Meanwhile, the states are kept almost undisturbed!Gentle Measurement Lemma
(Winter ’99)
Note: one-way LOCC measurement (1-LOCC) can be replaced by one side local measurement (1-LM)
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A technical lemma
Lemma
Note: one-way LOCC measurement (1-LOCC) can be replaced by one side local measurement (1-LM)
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Proof of Theorem 1------Monogamy relation for relative entropy of entanglement
To show
Proof:
(monotonicity)
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Proof of Theorem 1------Monogamy relation for relative entropy of entanglement
(joint convexity)
(1-LOCC of Tn)
(lemma & asym. cont.)
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Proof of Theorem 2------Commensurate lower bound for squashed entanglement
To show
Proof:(Theorem 1)
(Brandão, Christandl, Yard ’11, Lemma 1)
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Last remark
1-LM
One-way LOCC measurement can be replaced by one-side local measurement, which is a measurement on system A and an identity operation on system B.
Inspired by the results of (Brandão & Harrow ’12)
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Open questions
Applications of our results?
Faithfulness of multipartite squashed entanglement?
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Thank you!