1. A generalization of quantum Stein’s Lemma Fernando G.S.L. Brandão and Martin B. Plenio Tohoku University, 13/09/2008

2. Given n copies of a quantum state, with the promise that it is described either by or, determine the identity of the state. Measure two outcome POVM. Error probabilities - Type I error: - Type II error: (i.i.d.) Quantum Hypothesis Testing Null hypothesis Alternative hypothesis

3. Given n copies of a quantum state, with the promise that it is described either by or, determine the identity of the state. Measure two outcome POVM. Error probabilities - Type I error: - Type II error: (i.i.d.) Quantum Hypothesis Testing The state is

4. Given n copies of a quantum state, with the promise that it is described either by or, determine the identity of the state Measure two outcome POVM Error probabilities - Type I error: - Type II error: (i.i.d.) Quantum Hypothesis Testing

5. Several possible settings, depending on the constraints imposed on the probabilities of error E.g. in symmetric hypothesis testing, (i.i.d.) Quantum Hypothesis Testing Quantum Chernoff bound(Audenaert, Nussbaum, Szkola, Verstraete 07)

6. Several possible settings, depending on the constraints imposed on the probabilities of error E.g. in symmetric hypothesis testing, (i.i.d.) Quantum Hypothesis Testing Quantum Chernoff bound(Audenaert, Nussbaum, Szkola, Verstraete 07)

7. Several possible settings, depending on the constraints imposed on the probabilities of error E.g. in symmetric hypothesis testing, (i.i.d.) Quantum Hypothesis Testing Quantum Chernoff bound(Audenaert, Nussbaum, Szkola, Verstraete 07)

8. Asymmetric hypothesis testing Quantum Stein’s Lemma Quantum Stein ’ s Lemma (Hiai and Petz 91; Ogawa and Nagaoka 00)

9. Asymmetric hypothesis testing Quantum Stein’s Lemma Quantum Stein ’ s Lemma (Hiai and Petz 91; Ogawa and Nagaoka 00)

10. Most general setting - Null hypothesis (null): - Alternative hypothesis (alt.): Quantum Stein ’ s Lemma

11. Known results - (null) versus (alt.) Quantum Stein ’ s Lemma

12. Known results - (null) versus (alt.) Quantum Stein ’ s Lemma

13. Known results - (null) versus (alt.) Quantum Stein ’ s Lemma (Hayashi 00; Bjelakovic et al 04)

14. Known results - (null) versus (alt.) - Ergodic null hypothesis versus i.i.d. alternative hypothesis Quantum Stein ’ s Lemma (Hayashi 00; Bjelakovic et al 04) (Hiai and Petz 91)

15. Known results - (null) versus (alt.) - Ergodic null hypothesis versus i.i.d. alternative hypothesis - General sequence of states: Information spectrum Quantum Stein ’ s Lemma (Hayashi 00; Bjelakovic et al 04) (Hiai and Petz 91) (Han and Verdu 94; Nagaoka and Hayashi 07)

16. What about allowing the alternative hypothesis to be non-i.i.d. and to vary over a family of states? Only ergodicity and related concepts seems not to be enough to define a rate for the decay of Quantum Stein ’ s Lemma (Shields 93)

17. What about allowing the alternative hypothesis to be non-i.i.d. and to vary over a family of states? Only ergodicity and related concepts seems not to be enough to define a rate for the decay of Quantum Stein ’ s Lemma (Shields 93) This talk: A setting where the optimal rate can be determined for varying correlated alternative hypothesis

18. Consider the following two hypothesis - Null hypothesis: For every we have - Alternative hypothesis: For every we have an unknown state, where satisfies: 1. Each is closed and convex 2. Each contains the maximally mixed state 3. If, then 4. If and, then 5. If, then A generalization of Quantum Stein ’ s Lemma

19. Consider the following two hypothesis - Null hypothesis: For every we have - Alternative hypothesis: For every we have an unknown state, where satisfies 1. Each is closed and convex 2. Each contains the maximally mixed state 3. If, then 4. If and, then 5. If, then A generalization of Quantum Stein ’ s Lemma

20. Consider the following two hypothesis - Null hypothesis: For every we have - Alternative hypothesis: For every we have an unknown state, where satisfies 1. Each is closed and convex 2. Each contains the maximally mixed state 3. If, then 4. If and, then 5. If, then A generalization of Quantum Stein ’ s Lemma

21. theorem: Given satisfying properties 1-5 and, - (Direct Part) there is a s.t. A generalization of Quantum Stein ’ s Lemma

22. theorem: Given satisfying properties 1-5 and, - (Strong Converse) s.t. A generalization of Quantum Stein ’ s Lemma

23. We say is separable if If it cannot be written in this form, it is entangled The sets of separable sates over, satisfy properties 1-5 The rate function of the theorem is a well-known entanglement measure, the regularized relative entropy of entanglement A motivation: Entanglement theory (Vedral and Plenio 98)

24. Given an entangled state The theorem gives an operational interpretation to this measure as the optimal rate of discrimination of an entangled state to a arbitrary family of separable states More on the relative entropy of entanglement on Wednesday Regularized relative entropy of entanglement (Vedral and Plenio 98)

25. Cor: For every entangled state Regularized relative entropy of entanglement

26. Rate of conversion of two states by local operations and classical communication: The corollary implies that if is entangled, The mathematical definition of entanglement is equal to the operational: multipartite bound entanglement is real For bipartite systems see Yang, Horodecki, Horodecki, Synak-Radtke 05 Regularized relative entropy of entanglement

27. Asymptotic continuity: Let Non-lockability: Let Some elements of the proofs (Horodecki and Synak-Radtke 05; Christandl 06) (Horodecki 3 and Oppenheim 05)

28. Lemma: Let and s.t. Then s.t. and Lemma: Let, Some elements of the proofs (Datta and Renner 08) (Ogawa and Nagaoka 00)

29. Almost power states: Exponential de Finetti theorem: For any permutation- symmetric state there exists a measure over and states s.t. Some elements of the proofs (Renner 05)

30. Almost power states: Exponential de Finetti theorem: For any permutation- symmetric state there exists a measure over and states s.t. Some elements of the proofs (Renner 05)

31. (Proof sketch) We can write the statement of the theorem as The dual formulation of the convex optimization above reads It is then clear that it suffices to prove Elements of the proof

32. (Proof sketch) We can write the statement of the theorem as The dual formulation of the convex optimization above reads It is then clear that it suffices to prove Elements of the proof

33. (Proof sketch) We first show that for every Take sufficiently large such that Let be such that By the strong converse of quantum Stein’s Lemma As we find Elements of the proof

34. (Proof sketch) We first show that for every Take sufficiently large such that Let be such that By the strong converse of quantum Stein’s Lemma As we find Elements of the proof

35. (Proof sketch) We first show that for every Take sufficiently large such that Let be such that By the strong converse of quantum Stein’s Lemma As we find Elements of the proof

36. (Proof sketch) We now show Let be an optimal sequence in the eq. above We can write Assuming conversely that the limit is zero, we find Elements of the proof

37. (Proof sketch) We now show Let be an optimal sequence in the eq. above We can write Assuming conversely that the limit is zero, we find Elements of the proof

38. (Proof sketch) We now show Let be an optimal sequence in the eq. above We can write Assuming conversely that the limit is zero, we find Elements of the proof

39. (Proof sketch) We now show Let be an optimal sequence in the eq. above We can write Assuming conversely that the limit is zero, we find Then Elements of the proof

40. (Proof sketch) We now show Let be an optimal sequence in the eq. above We can write Assuming conversely that the limit is zero, we find Then Elements of the proof

41. (Proof sketch) Finally we now show with. Suppose conversely that.From we can write Note that we can take to be permutation-symmetric Elements of the proof

42. (Proof sketch) Define We have We can write Elements of the proof

43. (Proof sketch) Define We have We can write Elements of the proof

44. (Proof sketch) Because Therefore we can write and with

45. Elements of the proof (Proof sketch) Because Therefore we can write and with

46. Elements of the proof (Proof sketch) Because Therefore we can write and with

47. Elements of the proof (Proof sketch) Therefore with Finally

48. Elements of the proof (Proof sketch) Because Therefore we can write and with