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- 1. Relative entropy and squashed entanglement Ke Li (CQT, NUS) Andreas Winter (Universitat Autònoma de Barcelona) QIP2013 Beijing
- 2. Entanglement and entanglement measures , ,c d dE E K Entanglement is an important concept and resource. It is the 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
- 3. 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)
- 4. Post-measurement relative entropy of entanglement, with respect to restricted measurement classes (Piani ’09) Here M is a class of measurement, such as
- 5. 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)
- 6. 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
- 7. 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-norm bound: where ABρ( ) 0sq ABE ρ > ⇔
- 8. 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: )
- 9. 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:
- 10. 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 ,
- 11. Comparisons between entanglement measures We are mainly interested in two families of entanglement measures: Squashed-like measures { } Er 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) ,
- 12. Comparisons between entanglement measures (focusing mainly on regularized versions) We obtain: Previously known ones:
- 13. Proofs of Theorem 1 and Theorem 2
- 14. Quantum hypothesis testing with one-way LOCC operations Consider the setting of hypothesis testing two hypotheses: or tests allowed: on ; 1-LOCC implementable two errors: ,
- 15. 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)
- 16. A technical lemma Lemma Note: one-way LOCC measurement (1-LOCC) can be replaced by one side local measurement (1-LM)
- 17. Proof of Theorem 1------ Monogamy relation for relative entropy of entanglement To show Proof: (monotonicity)
- 18. Proof of Theorem 1------ Monogamy relation for relative entropy of entanglement (joint convexity) (1-LOCC of Tn) (lemma & asym. cont.)
- 19. Proof of Theorem 2------ Commensurate lower bound for squashed entanglement To show Proof: (Theorem 1) (Brandão, Christandl, Yard ’11, Lemma 1)
- 20. 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)
- 21. Open questions Applications of our results? Faithfulness of multipartite squashed entanglement?
- 22. Thank you!

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