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1. Fernando G.S.L. Brandão ETH Zürich Based on joint work with M. Christandl and J. Yard JourneesDeferation de Reserche en Mathematiques de Paris Centre/GT InformatiqueQuantique Paris, 09/05/2012 The Complexity of Quantum Entanglement

2. Quadratic vs Biquadratic Optimization • Problem 1: For M in H(Cd) (d x d matrix) compute • Very Easy! • Problem 2: For M in H(CdCl), compute This talk: Best known algorithm (and best hardness result) using ideas from Quantum Information Theory

3. Quadratic vs Biquadratic Optimization • Problem 1: For M in H(Cd) (d x d matrix) compute • Very Easy! • Problem 2: For M in H(CdCl), compute This talk: Best known algorithm (and best hardness result) using ideas from Quantum Information Theory

4. Quadratic vs Biquadratic Optimization • Problem 1: For M in H(Cd) (d x d matrix) compute • Very Easy! • Problem 2: For M in H(CdCl), compute This talk: Best known algorithm (and best hardness result) using ideas from Quantum Information Theory

5. Outline • The Problem • Quantum States • Quantum Entanglement • The Algorithm • Parrilo-Lasserre Relaxation • Monogamy of Entanglement • Quantum de Finetti Theorem • Applications • A new characterization of Quantum NP • Small Set Expansion • Proof Ideas

6. Quantum States • Pure States: norm-one vector in Cd: • Mixed States: positive semidefinite matrix of unit trace: Dirac notation reminder:

7. Quantum Measurements • To any experiment with d outcomes we associate d • positive matrices {Mk} such that • and calculate probabilities as • E.g. For pure states,

8. Quantum Entanglement • Pure States: • If , it’s separable • otherwise, it’s entangled. • Mixed States: • If it’s separable • otherwise, it’s entangled.

9. Quantum Entanglement • Pure States: • If , it’s separable • otherwise, it’s entangled. • Mixed States: • If , it’s separable • otherwise, it’s entangled.

10. A Physical Definition of Entanglement LOCC: Local quantum Operations and Classical Communication Separable states can be created by LOCC: Entangled states cannot be created by LOCC: non-classicalcorrelations

11. The Separability Problem • Given • is it separable or entangled? • (Weak Membership: WSEP(ε, ||*||) Given ρAB • determine if it is separable, or ε-way from SEP D SEP

12. The Problem (for experimentalists)

13. The Problem (for experimentalists)

14. The Problem (for experimentalists)

15. Relevance • Quantum Cryptography • Security only if state is entangled • Quantum Communication • Advantage over classical (e.g. teleportation, • dense coding) only if state is entangled • Computational Physics • Entanglement responsible for difficulty of • simulation of quantum systems

16. Deciding Entanglement • The problem of deciding whether a state is entangled • has been considered since the early days of the field of quantum information theory • is regarded as a computationally difficult problem • In this talk I’ll discuss the fastest known algorithm for • this problem

17. The Separability Problem (again) • Given • is it separable or entangled? • (Weak Membership: WSEP(ε, ||*||) Given ρAB • determine if it is separable, or ε-way from SEP D SEP

18. The Separability Problem (again) • Given • is it separable or entangled? • (Weak Membership: WSEP(ε, ||*||) Given ρAB • determine if it is separable, or ε-way from SEP Which norm should we use? D SEP

19. Norms on Quantum States • How to quantify the distance in Weak-Membership? • Euclidean Norm (Hilber-Schmidt): • ||X||2 = tr(XTX)1/2 • Trace Norm • ||X||1 = tr((XTX)1/2) • Obs: ||X||1 ≥||X||2≥d-1/2||X||1

20. The LOCC Norm • Operational interpretation trace norm: • ||ρ – σ||1= 2 max 0<M<Itr(M(ρ – σ)) • optimal bias of distinguishing the two states by quantum measurements • For ρAB, σAB define • ||ρ – σ||LOCC= 2 max 0<M<Itr(M(ρ – σ)) : {M, I - M} in LOCC LOCC: Local quantum Operations and Classical Communication

21. The LOCC Norm • Operational interpretation trace norm: • ||ρ – σ||1= 2 max 0<M<Itr(M(ρ – σ)) • optimal bias of distinguishing the two states by quantum measurements • For ρAB, σAB define the LOCC norm • ||ρ – σ||LOCC= 2 max 0<M<Itr(M(ρ – σ)) : {M, I - M} in LOCC • Optimal bias of distinguishing two states by LOCC measurements • E.g. (one-way LOCC)

22. Optimization Over Separable States (Best Separable State BSS(ε)) Given estimate to additive error ε

23. Previous Work When is ρAB entangled? - Decide if ρAB is separable or ε-away from separable Beautiful theory behind it (PPT, entanglement witnesses, etc) Horribly expensive algorithms State-of-the-art: 2O(|A|log|B|log (1/ε))time complexity for either ||*||2 or ||*||1 norms (Doherty, Parrilo, Spedalieri ‘04)

24. Hardness Results When is ρAB entangled? - Decide if ρAB is separable or ε-away from separable (Gurvits ‘02) NP-hard with ε=1/exp(|A||B|) (Gharibian ‘08, Beigi ‘08) NP-hard with ε=1/poly((|A||B|)1/2) (Beigi&Shor ‘08) Favorite separability tests fail (Harrow&Montanaro ‘10) No exp(O(|A|1-ν|A|1-μ))time algorithm for membership in any convex set within ε=Ω(1) trace distance to SEP and any ν+μ>0, unless ETH fails ETH (Exponential Time Hypothesis): SAT cannot be solved in 2o(n) time (Impagliazzo&Paruti’99)

25. Hardness Results When is ρAB entangled? - Decide if ρAB is separable or ε-away from separable (Gurvits ‘02) NP-hard with ε=1/exp(|A||B|) (Gharibian ‘08, Beigi ‘08) NP-hard with ε=1/poly(|A||B|) (Beigi&Shor ‘08) Favorite separability tests fail (Harrow&Montanaro ‘10) No exp(O(|A|1-ν|A|1-μ))time algorithm for membership in any convex set within ε=Ω(1) trace distance to SEP and any ν+μ>0, unless ETH fails ETH (Exponential Time Hypothesis): SAT cannot be solved in 2o(n) time (Impagliazzo&Paruti’99)

26. Hardness Results When is ρAB entangled? - Decide if ρAB is separable or ε-away from separable (Gurvits ‘02) NP-hard with ε=1/exp(|A||B|) (Gharibian ‘08, Beigi ‘08) NP-hard with ε=1/poly(|A||B|) (Beigi, Shor ‘08) Favorite separability tests fail (Harrow&Montanaro ‘10) No exp(O(|A|1-ν|A|1-μ))time algorithm for membership in any convex set within ε=Ω(1) trace distance to SEP and any ν+μ>0, unless ETH fails ETH (Exponential Time Hypothesis): SAT cannot be solved in 2o(n) time (Impagliazzo&Paruti’99)

27. Hardness Results When is ρAB entangled? - Decide if ρAB is separable or ε-away from separable (Gurvits ‘02) NP-hard with ε=1/exp(|A||B|) (Gharibian ‘08, Beigi ‘08) NP-hard with ε=1/poly(|A||B|) (Beigi, Shor ‘08) Favorite separability tests fail (Harrow, Montanaro ‘10) No exp(O(log1-ν|A|log1-μ|B|)) time algorithm for membership in any convex set within ε=Ω(1) trace distance to SEP, and any ν+μ>0, unless ETH fails ETH (Exponential Time Hypothesis): SAT cannot be solved in 2o(n) time (Impagliazzo&Paruti’99)

28. Algorithms for BSS Estimate with additive error ε State-of-the-art: 2O((|A|+|B|)log (1/ε))time complexity Exhaustive search over ε-nets on A and B!

29. Hardness Results for BSS Estimate with additive error ε (Gurvits ‘02, Gharibian ‘08, Beigi ‘08) NP-hard with ε=1/poly(|A||B|) (Harrow, Montanaro’10, built on Aaronson et al ‘08) No exp(O(log1-ν|A|log1-μ|B|||M||∞)) time algorithm for any ν+μ>0 and constant ε, unless ETH fails

30. Main Result 1: Weak Membership (B., Christandl, Yard ‘10) There is aexp(O(ε-2log|A|log|B|)) time algorithm for WSEP(||*||, ε) (in ||*||2 or ||*|LOCC)

31. Main Result 1: Weak Membership (B., Christandl, Yard ‘10) There is aexp(O(ε-2log|A|log|B|)) time algorithm for WSEP(||*||, ε) (in ||*||2 or ||*|LOCC) Remind: NP-hard for ε = 1/poly(|A||B|) in ||*||2 (Gurvits ‘02, Gharibian ‘08, Beigi ‘08) Corollary: the problem in ||*||2is not NP-hard forε = 1/polylog(|A||B|), unless ETH fails

32. Main Result 2: Best Separable State • (BCY ‘10) • There is aexp(O(ε-2log|A|log|B|(||M||2)2)) time algorithm for BSS(ε) • For M in LOCC, there is aexp(O(ε-2log|A|log|B|)) time algorithm for BSS(ε)

33. Main Result 2: Best Separable State • (BCY ‘10) • There is aexp(O(ε-2log|A|log|B|(||M||2)2)) time algorithm for BSS(ε) • For M in LOCC, there is aexp(O(ε-2log|A|log|B|)) time algorithm for BSS(ε) Contrast with: (Harrow, Montanaro ‘10) No exp(O(log1-ν|A|log1-μ|B|||M||∞)) time algorithm for any ν+μ>0 and constant ε, unless ETH fails, even for separable M: . Remember: Part 2 works for

34. Main Result 2: Best Separable State Quantum Info Remark: The difficulty to show optimality of the algorithm is the existence of separable measurements that are not LOCC, a well studied phenomena in quantum information (e.g. Bennett et al ‘98). Here we have a new computational-complexity motivation for further studying the problem! • (BCY ‘10) • There is aexp(O(ε-2log|A|log|B|(||M||2)2)) time algorithm for BSS(ε) • For M in LOCC, there is aexp(O(ε-2log|A|log|B|)) time algorithm for BSS(ε) Contrast with: (Harrow, Montanaro ‘10) No exp(O(log1-ν|A|log1-μ|B|||M||∞)) time algorithm for any ν+μ>0 and constant ε, unless ETH fails, even for separable M: . Remember: Part 2 works for

35. The Algorithm • We consider the a Parrilo-Lasserre hierarchy of SDP relaxations to the problem introduced in (Doherty, Parrilo and Spedalieri’01) • We prove it converges to a good approximate solution in a O(log|B|) number of rounds. Previously convergence only in Ω(|B|) rounds • was known.

36. Optimization Over Separable States (again) (Best Separable State BSS(ε)) Given estimate to additive error ε This is a polynomial optimization problem. One can calculate a sequence of SDP approximations to it following the approach of (Parrilo ‘00, Lasserre’01) We’ll derive the SDP hierachy by a quantum argument

37. Entanglement Monogamy Classical correlations are shareable: Given separable state Consider the symmetric extension B1 B2 B3 B4 A … Bk Def. ρAB is k-extendible if there is ρAB1…Bk s.t for all j in [k], tr\ Bj(ρAB1…Bk) = ρAB

38. 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

39. Entanglement Monogamy Quantum correlations are non-shareable: ρAB separable iffρABk-extendible for all k Follows from: Quantum de Finetti Theorem (Stormer’69, Hudson & Moody ’76, Raggio & Werner ’89) Monogamy of entanglement: Very useful concept in general, application e.g. in quantum key distribution

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

41. 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 k-ext state ρABs.t. For other norms (||*||2, ||*||LOCC, …) no better bound known.

42. Exponentially Improved de Finetti type bound (B., Christandl, Yard ‘10) For any k-extendible ρAB, with||*|| equals ||*||2 or ||*||LOCC Bound proportional to the (square root) of thenumber of qubits:exponential improvement over previous bound

43. How long does it take to check if a k-extension exists? • Search for a symmetric extension is a semidefinite program (Doherty, Parrilo, Spedalieri ‘04) • Can be solved in poly(n) time in the number of variables n • n = |A|2|B|2k • Our bound implies k = O(ε-2log|A|) • Time Complexity: • poly(|A||B|2k) = exp(O(ε-2log|A|log|B|))

44. Does it work for 1-norm? • There are k-extendible states s.t. • For such states the SDP hierarchy only gives good solutions for k = O(|B|), which requires exponential time • But we know also: • So, hard instances are always “data hiding” states, i.e.

45. Does it work for 1-norm? • There are k-extendible states s.t. • For such states the SDP hierarchy only gives good solutions for k = O(|B|), which requires exponential time • But we know also: • So, hard instances are always “data hiding” states, i.e.

46. Does it work for 1-norm? • There are k-extendible states s.t. • For such states the SDP hierarchy only gives good solutions for k = O(|B|), which requires exponential time • But we know also: • So, hard instances are always “data hiding” states, i.e.

47. Does it work for 1-norm? • There are k-extendible states s.t. • For such states the SDP hierarchy only gives good solutions for k = O(|B|), which requires exponential time • But we know also: • So, hard instances are always “data hiding” states, i.e.

48. Algorithm for Best Separable State The idea Optimize over k=O(log|A|ε-2 (||X||2)2) extension of ρAB by SDP This is precisely the Parrilo-Lasserre hierarchy for the problem! (written in a somewhat different form) By Cauchy Schwartz: By de Finetti Bound:

49. Application 1: Quantum NP
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