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Faithful Squashed Entanglement

Fernando G.S.L. Brand ão 1 Matthias Christandl 2 Jon Yard 3 1. Universidade Federal de Minas Gerais , Brazil 2. ETH Zürich, Switzerland 3. Los Alamos Laboratory, USA. Faithful Squashed Entanglement. w ith applications to separability testing and quantum Merlin-Arthur games.

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Faithful Squashed Entanglement

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  1. Fernando G.S.L. Brandão1 Matthias Christandl2 Jon Yard3 1. Universidade Federal de Minas Gerais, Brazil 2. ETH Zürich, Switzerland 3. Los Alamos Laboratory, USA Faithful Squashed Entanglement with applications to separability testing and quantum Merlin-Arthur games

  2. Mutual Information vs Conditional Mutual Information Mutual Information:Measures the correlations of A and B in ρAB I(A:B)ρ := S(A)ρ + S(B)ρ – S(AB)ρ

  3. Mutual Information vs Conditional Mutual Information Mutual Information:Measures the correlations of A and B in ρAB I(A:B)ρ := S(A)ρ + S(B)ρ – S(AB)ρ Always positive: I(A:B)ρ ≥ 0 (subadditivity of entropy)

  4. Mutual Information vs Conditional Mutual Information Mutual Information:Measures the correlations of A and B in ρAB I(A:B)ρ := S(A)ρ + S(B)ρ – S(AB)ρ Always positive: I(A:B)ρ ≥ 0 (subadditivity of entropy) When does it vanish? I(A:B)ρ = 0 iffρAB = ρAρB

  5. Mutual Information vs Conditional Mutual Information Mutual Information:Measures the correlations of A and B in ρAB I(A:B)ρ := S(A)ρ + S(B)ρ – S(AB)ρ Always positive: I(A:B)ρ ≥ 0 (subadditivity of entropy) When does it vanish? I(A:B)ρ = 0 iffρAB = ρAρB Approximate version? Pinsker’s inequality: Remark: dimension-independent! Useful in many application in QIT (e.g. decoupling, QKD, …)

  6. Mutual Information vs Conditional Mutual Information Conditional Mutual Information:Measures the correlations of A and B relative to Ein ρABE I(A:B|E)ρ := S(AE)ρ + S(BE)ρ – S(ABE)ρ – S(E)ρ

  7. Mutual Information vs Conditional Mutual Information Conditional Mutual Information:Measures the correlations of A and B relative to Ein ρABE I(A:B|E)ρ := S(AE)ρ + S(BE)ρ – S(ABE)ρ – S(E)ρ Always positive: I(A:B|E)ρ ≥ 0 (strong-subadditivity of entropy) (Lieb, Ruskai ‘73)

  8. Mutual Information vs Conditional Mutual Information Conditional Mutual Information:Measures the correlations of A and B relative to Ein ρABE I(A:B|E)ρ := S(AE)ρ + S(BE)ρ – S(ABE)ρ – S(E)ρ Always positive: I(A:B|E)ρ ≥ 0 (strong-subadditivity of entropy) When does it vanish? I(A:B|E)ρ = 0 iffρABEis a “Quantum Markov Chain State” E.g. (Lieb, Ruskai ‘73) (Hayden, Jozsa, Petz, Winter ‘04)

  9. Mutual Information vs Conditional Mutual Information Conditional Mutual Information:Measures the correlations of A and B relative to Ein ρABE I(A:B|E)ρ := S(AE)ρ + S(BE)ρ – S(ABE)ρ – S(E)ρ Always positive: I(A:B|E)ρ ≥ 0 (strong-subadditivity of entropy) When does it vanish? I(A:B|E)ρ = 0 iffρABEis a “Quantum Markov Chain State” E.g. Approximate version??? … (Lieb, Ruskai ‘73) (Hayden, Jozsa, Petz, Winter ‘04)

  10. Outline • I(A:B|E)≈0 characterization • Applications: • Squashed Entanglement • de Finetti-type bounds • Algorithm for Separability • A new characterization of QMA • Proof

  11. No-Go For Approximate Version A naïve guess for approximate version (à la Pinsker):

  12. No-Go For Approximate Version A naïve guess for approximate version (à la Pinsker): = = It fails badly! Ω(1) O(|A|-1) (Christandl, Schuch, Winter ’08) E.g. Antisymmetric Werner state

  13. Main Result Thm: (B., Christandl, Yard ’10)

  14. Main Result Thm: (B., Christandl, Yard ’10) (Euclidean norm or (one-way) LOCC norm)

  15. Main Result Thm: (B., Christandl, Yard ’10) (Euclidean norm or (one-way) LOCC norm) Pointed out yesterday by David Reeb

  16. Main Result Thm: (B., Christandl, Yard ’10) (Euclidean norm or (one-way) LOCC norm) The Euclidean (Frobenius)norm: ||X||2 = tr(XTX)1/2

  17. Main Result Thm: (B., Christandl, Yard ’10) (Euclidean norm or (one-way) LOCC norm) Thetrace norm: ||X||1 = ½ + ½ max0≤A≤I|tr(AX)| ||ρ-σ||1: optimal bias

  18. Main Result Thm: (B., Christandl, Yard ’10) (Euclidean norm or (one-way) LOCC norm) The LOCC norm: ||X||LOCC = ½ + ½ max0≤A≤I|tr(AX)| : {A, I-A} in LOCC ||ρ-σ||LOCC: optimal bias by LOCC

  19. Main Result Thm: (B., Christandl, Yard ’10) (Euclidean norm or (one-way) LOCC norm) The one-way LOCC norm: ||X||LOCC(1) = ½ + ½ max0≤A≤I|tr(AX)| : {A, I-A} in one-way LOCC ||ρ-σ||LOCC: optimal bias by one-way LOCC

  20. The Power of LOCC Thm: (B., Christandl, Yard ’10) (Euclidean norm or (one-way) LOCC norm) (Matthews, Wehner, Winter ‘09) For X in A B Interesting one, uses a covariant random local measurement

  21. Squashed Entanglement (Christandl, Winter ‘04) Squashed entanglement: Esq(ρAB) = infπ{ ½ I(A:B|E)π : trE(πABE) = ρAB} Open question:Is it faithful? i.e. Is Esq(ρAB) > 0 for every entangled ρAB?

  22. Squashed Entanglement (Christandl, Winter ‘04) Squashed entanglement: Esq(ρAB) = infπ{ ½ I(A:B|E)π : trE(πABE) = ρAB} Open question:Is it faithful? i.e. Is Esq(ρAB) > 0 for every entangled ρAB? Corollary:

  23. Squashed Entanglement (Christandl, Winter ‘04) Squashed entanglement: Esq(ρAB) = infπ{ ½ I(A:B|E)π : trE(πABE) = ρAB} Corollary Proof: From Follows: General two-way LOCC: monotonicity of squashed entanglement under LOCC

  24. Entanglement Zoo

  25. Entanglement Zoo

  26. Entanglement Monogamy Classical correlations are shareable: B1 B2 B3 Def. ρAB is k-extendible if there is ρAB1…Bk s.t for all j in [k] tr\ Bj(ρAB1…Bk) = ρAB B4 A … Bk Separable states are k-extendible for every k.

  27. Entanglement Monogamy Quantum correlations are non-shareable: ρABentanglediffρABnot k-extendible for some k - Follows from: Quantum de Finetti Theorem (Stormer’69, Hudson & Moody ’76, Raggio & Werner ’89) E.g. - Any pure entangled state is not 2-extendible - The d x d antisymmetric state is not d-extendible

  28. Entanglement Monogamy Quantitative version: For any k-extendible ρAB, - Follows from: finite quantum de Finetti Theorem (Christandl, König, Mitchson, Renner ‘05)

  29. Entanglement Monogamy Quantitative version: For any k-extendible ρAB, - Follows from: finite quantum de Finetti Theorem (Christandl, König, Mitchson, Renner ‘05) Close to optimal: there is a state ρABs.t. (guess which? ) For other norms (||*||2, ||*||LOCC, …) no better bound known.

  30. Exponentially Improved de Finetti type bound Corollary For any k-extendible ρAB, with||*|| equals ||*||2 or ||*||LOCC Bound proportional to the (square root) of the number of qubits: exponential improvement over previous bound

  31. Exponentially Improved de Finetti type bound Corollary For any k-extendible ρAB, with||*|| equals ||*||2 or ||*||LOCC Proof: Esq satisfies monogamy relation (Koashi, Winter ’05) For ρABk-extendible:

  32. Exponentially Improved de Finetti type bound Corollary For any k-extendible ρAB, with||*|| equals ||*||2 or ||*||LOCC (Close-to-Optimal) There is k-extendible state ρABs.t.

  33. Exponentially Improved de Finetti type bound

  34. The separability problem When is ρAB entangled? - Decide if ρAB is separable or ε-away from separable Beautiful theory behind it (PPT, entanglement witnesses, symmetric extensions, etc) Horribly expensive algorithms State-of-the-art: 2O(|A|log(1/ε))time complexity (Doherty, Parrilo, Spedalieri ‘04)

  35. The separability problem When is ρAB entangled? - Decide if ρAB is separable or ε-away from separable Hardness results: (Gurvits ‘02) NP-hard with ε=1/exp(d) (d:= (|A||B|)1/2) (Gharibian ‘08, Beigi ‘08) NP-hard with ε=1/poly(d) (Beigi&Shor ‘08) Favorite separability tests fail (Harrow&Montanaro ‘10) No exp(O(d2-δ))time algorithm for membership in any convex set within ε=Ω(1) trace distance to SEP, unless ETH fails ETH (Exponential Time Hypothesis): SAT cannot be solved in 2o(n) time (Impagliazzo&Paruti’99)

  36. Quasi-polynomial Algorithm Corollary There is aexp(O(ε-2log|A|log|B|)) time algorithm for deciding separability (in ||*||2 or ||*||LOCC)

  37. Quasi-polynomial Algorithm Corollary There is aexp(O(ε-2log|A|log|B|)) time algorithm for deciding separability (in ||*||2 or ||*||LOCC) The idea (Doherty, Parrilo, Spedalieri ‘04) Search for a k=O(log|A|/ε2) extension of ρAB by SDP Complexity SDP of size |A|2|B|2k = exp(O(ε-2log|A|log|B|))

  38. Quasi-polynomial Algorithm Corollary There is aexp(O(ε-2log|A|log|B|)) time algorithm for deciding separability (in ||*||2 or ||*||LOCC) NP-hardness for ε = 1/poly(d) is shown using ||*||2 From corollary: the problem in ||*||2cannot be NP-hard forε = 1/polylog(d), unless ETH fails

  39. Best Separable State Problem BSS(ε) Problem: Given X, approximate to additive error ε Corollary There is aexp(O(ε-2 log|A| log|B| (||X||2)2)) time algorithm for BSS(ε)

  40. Best Separable State Problem BSS(ε) Problem: Given X, approximate to additive error ε Corollary There is aexp(O(ε-2 log|A| log|B| (||X||2)2)) time algorithm for BSS(ε) The idea Optimize over k=O(log|A|ε-2 (||X||2)2) extension of ρAB by SDP

  41. Best Separable State Problem BSS(ε) Problem: Given X, approximate to additive error ε Corollary There is aexp(O(ε-2 log|A| log|B| (||X||2)2)) time algorithm for BSS(ε) (Harrow and Montanaro ‘10): BSS(ε) for ε=Ω(1) and ||X||∞ ≤ 1 cannot be solved in exp(O(log1-ν|A| log1-μ|B|)) time for any ν + μ > 0 unless ETH fails

  42. QMA

  43. QMA

  44. QMA - Quantum analogue of NP (or MA) - Local Hamiltonian Problem, … Is QMA a robust complexity class? (Aharonov, Regev ‘03) superverifiers doesn’t help (Marriott, Watrous ‘05) Exponential amplification with fixed proof size (Beigi, Shor, Watrous ‘09) logarithmic size interaction doesn’t help

  45. New Characterization QMA Corollary QMA doesn’t change allowing k = O(1) different proofs if the verifier can only apply LOCC measurements in the k proofs

  46. New Characterization QMA Corollary QMA doesn’t change allowing k = O(1) different proofs if the verifier can only apply LOCC measurements in the k proofs DefQMAm,s,c(k): analogue of QMA with k proofs, proof size m, soundness sand completeness c.

  47. New Characterization QMA Corollary QMA doesn’t change allowing k = O(1) different proofs if the verifier can only apply LOCC measurements in the k proofs DefQMAm,s,c(k): analogue of QMA with k proofs, proof size m, soundness sand completeness c. DefLOCCQMAm,s,c(k): analogue of QMA with k proofs, proof size m, soundness s, completeness c and LOCC verification procedure along the k proofs.

  48. New Characterization QMA Corollary QMA = LOCCQMA(k), k = O(1) LOCCQMAm,s,c(2) = QMAO(m2ε-2),s+ε,c Contrast: QMAm,s,c(2) not in QMAO(m2-δε-2),s+ε,c for ε = O(1) and δ>0 unless Quantum ETH*fails Follows from Harrow and Montanaro ‘10 (based on Aaroson et al ‘08) * Quantum ETH: SAT cannot be solved in 2o(n) quantum time

  49. New Characterization QMA Corollary QMA = LOCCQMA(k), k = O(1) LOCCQMAm,s,c(2) = QMAO(m2ε-2),s+ε,c • Idea to simulate LOCCQMAm,s,c(2) in QMA: • Arthur asks for proof ρ on AB1B2…Bkwith k = mε-2 • He symmetrizes the B systems and apply the original verification prodedure to AB1 • Correcteness • de Finetti bound implies:

  50. Proof

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