Fernando G.S.L. Brand ão ETH Zürich Princeton EE Department, March 2013

1. Fernando G.S.L. Brandão ETH Zürich Princeton EE Department, March 2013 Monogamy of Quantum Entanglement

2. Simulating quantum is hard More than 25%of DoE supercomputer power is devoted to simulating quantum physics Can we get a better handle on this simulation problem?

3. Simulating quantum is hard, secrets are hard to conceal More than 25%of DoE supercomputer power is devoted to simulating quantum physics Can we get a better handle on this simulation problem? All current cryptography is based on unproven hardness assumptions Can we have better security guarantees for our secrets?

4. Quantum Information Science… … gives a path for solving both problems. But it’s a long journey QIS is at the crossover of computer science, engineering and physics Physics CS Engineering

5. The Two Holy Grails of QIS Quantum Computation: Use of engineered quantum systemsfor performing computation Exponential speed-ups over classical computing E.g. Factoring (RSA) Simulating quantum systems Quantum Cryptography: Use of engineered quantum systems for secret key distribution Unconditional security based solely on the correctness of quantum mechanics

6. The Two Holy Grails of QIS Quantum Computation: Use of well controlled quantum systemsfor performing computations. Exponential speed-ups over classical computing Quantum Cryptography: Use of engineered quantum systems for secret key distribution Unconditional security based solely on the correctness of quantum mechanics State-of-the-art: +-5 qubitscomputer Can prove (with high probability) that 15 = 3 x 5

7. The Two Holy Grails of QIS Quantum Computation: Use of well controlled quantum systemsfor performing computations. Exponential speed-ups over classical computing Quantum Cryptography: Use of well controlled quantum systems for secret key distribution Unconditional security based solely on the correctness of quantum mechanics State-of-the-art: +-5 qubitscomputer Can prove (with high probability) that 15 = 3 x 5 State-of-the-art: +-100 km Still many technological challenges

8. While waiting for a quantum computer …how can a theorist help? Answer 1: figuring out what to do with a quantum computer and the fastest way of building one (quantum communication, quantum algorithms, quantum fault tolerance, quantum simulators, …)

9. While waiting for a quantum computer …how can a theorist help? Answer 1: figuring out what to do with a quantum computer and the fastest way of building one (quantum communication, quantum algorithms, quantum fault tolerance, quantum simulators, …) Answer 2: finding applications of QISindependent of building a quantum computer

10. While waiting for a quantum computer …how can a theorist help? Answer 1: figuring out what to do with a quantum computer and the fastest way of building one (quantum communication, quantum algorithms, quantum fault tolerance, quantum simulators, …) Answer 2: finding applications of QISindependent of building a quantum computer This talk

11. Quantum Mechanics I probability theory based on L2 norm instead of L1 • States: norm-one vector in Cd: • Dynamics: L2 norm preserving linear maps, • unitary matrices: UUT = I • Observation: To any experiment with d outcomes we • associate dPSD matrices {Mk} such that ΣkMk=I • Born’s rule:

12. Quantum Mechanics II probability theory based on PSD matrices • Mixed States: PSD matrix of unit trace: • Dynamics: linear operations mapping density matrices to density matrices (unitaries + adding ancilla + ignoring subsystems) • Born’s rule: Dirac notation reminder:

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

14. 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

15. Quantum Features Superposition principle uncertainty principle interference non-locality

16. Monogamy of Entanglement

17. Monogamy of Entanglement Maximally Entangled State: Correlations of A and B in cannot be shared: If ρABE is is s.t.ρAB = , then ρABE= Only knowing that the quantum correlations of A and B are very strong one can infer A and B are not correlated with anything else. Purely quantum mechanical phenomenon!

18. Quantum Key Distribution… … as an application of entanglement monogamy (Bennett, Brassard 84, Ekert 91) • Using insecure channel Alice and Bob share entangled state ρAB • Applying LOCC they distill

19. Quantum Key Distribution… … as an application of entanglement monogamy (Bennett, Brassard 84, Ekert 91) • Using insecure channel Alice and Bob share entangled state ρAB • Applying LOCC they distill • Let πABE be the state of Alice, Bob and the Eavesdropper. • Since πAB= , πAB= . Measuring A and B • in the computational basis give 1 bit of secret key.

20. Outline Entanglement Monogamy How general is the concept? Can we make it quantitative? Are there other applications/implications? 1. Detecting Entanglement and Sum-Of-Squares Hierarchy 2. Accuracy of Mean-Field Approximation 3. Other Applications

21. 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 Next: Best known algorithm (and best hardness result) using ideas from Quantum Information Theory

22. 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 Next: Best known algorithm using monogamy of entanglement

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

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

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

26. The Separability Problem • Given • is it entangled? • (Weak Membership: WSEP(ε, ||*||)) Given ρAB • determine if it is separable, or ε-away from SEP • Dual Problem: Optimization over separable states

27. The Separability Problem • Given • is it entangled? • (Weak Membership: WSEP(ε, ||*||)) Given ρAB • determine if it is separable, or ε-away from SEP • Dual Problem: Optimization over separable states • Relevance: Entanglement is a resource in quantum • cryptography, quantum communication, etc…

28. 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|)time complexity Same for estimating hSEP(no better than exaustive search!) dim of A

29. Hardness Results When is ρAB entangled? - Decide if ρAB is separable or ε-away from separable (Gurvits’02, Gharibian’08, …) NP-hard with ε=1/poly(|A||B|) (Harrow, Montanaro ‘10) Noexp(O(log1-ν|A|log1-μ|B|)) time algorithm with ε=Ω(1) and any ν+μ>0 for unless ETH fails ETH (Exponential Time Hypothesis): SAT cannot be solved in 2o(n) time (Impagliazzo&Paruti’99)

30. Quasipolynomial-time Algorithm (B., Christandl, Yard ’11) There is an algorithm with run timeexp(O(ε-2log|A|log|B|)) for WSEP(||*||, ε)

31. Quasipolynomial-time Algorithm (B., Christandl, Yard ’11) There is an algorithm with run timeexp(O(ε-2log|A|log|B|)) for WSEP(||*||, ε) Corollary: Solving WSEP(||*||, ε) is not NP-hard for ε= 1/polylog(|A||B|), unless ETH fails Contrast with: (Gurvits’02, Gharibian ‘08) Solving WSEP(||*||, ε) NP-hard for ε = 1/poly(|A||B|)

32. Quasipolynomial-time Algorithm (B., Christandl, Yard ’11) For M in 1-LOCC, can computehSEP(M) within additive error ε in timeexp(O(ε-2log|A|log|B|)) M in 1-LOCC: Contrast with (Harrow, Montanaro’10) No exp(O(log1-ν|A|log1-μ|B|))algorithm for hSEP(M) with ε=Ω(1), for separable M M in SEP:

33. Algorithm: SoS Hierarchy • Polynomial optimization over hypersphere • Sum-Of-Squares (Parrilo/Lasserre) hierarchy: • gives sequence of SDPs that approximate hSEP(M) • - Round k takes time dim(M)O(k) • Converge to hSEP(M) when k -> ∞ • SoS is the strongest SDP hierarchy known for • polynomial optimization (connections with SoS proof system, • real algebraic geometric, Hilbert’s 17th problem, …) • We’ll derive SoS hierarchy by a quantum argument

34. Algorithm: SoS Hierarchy • Polynomial optimization over hypersphere • Sum-Of-Squares (Parrilo/Lasserre) hierarchy: • gives sequence of SDPs that approximate hSEP(M) • - Round k SDP has size dim(M)O(k) • Converge to hSEP(M) when k -> ∞

35. Algorithm: SoS Hierarchy • Polynomial optimization over hypersphere • Sum-Of-Squares (Parrilo/Lasserre) hierarchy: • gives sequence of SDPs that approximate hSEP(M) • - Round k SDP has size dim(M)O(k) • Converge to hSEP(M) when k -> ∞ • SoS is the strongest SDP hierarchy known for • polynomial optimization (connections with SoS proof system, • real algebraic geometric, Hilbert’s 17th problem, …) • We’ll derive SoS hierarchy exploring ent. monogamy

36. 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…Bks.t for all j in [k], tr\ Bj(ρAB1…Bk) = ρAB

37. Monogamy Defines Entanglement (Stormer’69, Hudson & Moody ’76, Raggio & Werner ’89) ρAB separable iffρAB is k-extendible for all k

38. SoS as optimization over k-extensions (Doherty, Parrilo, Spedalieri ‘01) k-level SoS SDP for hSEP(M) is equivalent to optimization over k-extendible states (plus PPT (positive partial transpose) test):

39. How close to separable are k-extendible states? (Koenig, Renner ‘04) De Finetti Bound If ρABis k-extendible Improved de Finetti Bound (B., Christandl, Yard ’11) IfρABis k-extendible:

40. How close to separable are k-extendible states? (Koenig, Renner ‘04) De Finetti Bound If ρABis k-extendible Improved de Finetti Bound (B., Christandl, Yard ’11) IfρABis k-extendible:

41. How close to separable are k-extendible states? Improved de FinettiBound IfρABis k-extendible: k=2ln(2)ε-2log|A| rounds of SoS solves WSEP(ε)with a SDP of size |A||B|k= exp(O(ε-2log|A| log|B|)) ε SEP D

42. Proving… Improved de FinettiBound IfρABis k-extendible: Proof is information-theoretic Mutual Information: I(A:B)ρ := H(A) + H(B) – H(AB) H(A)ρ:= -tr(ρ log(ρ))

43. Proving… Improved de FinettiBound IfρABis k-extendible: Proof is information-theoretic Mutual Information: I(A:B)ρ := H(A) + H(B) – H(AB) H(A)ρ:= -tr(ρ log(ρ)) Let ρAB1…Bkbe k-extension of ρAB log|A| > I(A:B1…Bk) = I(A:B1)+I(A:B2|B1)+…+I(A:Bk|B1…Bk-1) For some l<k: I(A:Bl|B1…Bl-1) < log|A|/k (chain rule)

44. Proving… Improved de FinettiBound IfρABis k-extendible: Proof is information-theoretic Mutual Information: I(A:B)ρ := H(A) + H(B) – H(AB) H(A)ρ:= -tr(ρ log(ρ)) Let ρAB1…Bkbe k-extension of ρAB log|A| > I(A:B1…Bk) = I(A:B1)+I(A:B2|B1)+…+I(A:Bk|B1…Bk-1) For some l<k: I(A:Bl|B1…Bl-1) < log|A|/k (chain rule) What does it imply?

45. Quantum Information? Nature isn't classical, dammit, and if you want to make a simulation of Nature, you'd better make it quantum mechanical, and by golly it's a wonderful problem, because it doesn't look so easy.

46. Quantum Information? Nature isn't classical, dammit, and if you want to make a simulation of Nature, you'd better make it quantum mechanical, and by golly it's a wonderful problem, because it doesn't look so easy. information theory • Bad news • Only definition I(A:B|C)=H(AC)+H(BC)-H(ABC)-H(C) • Can’t condition on quantum information. • I(A:B|C)ρ ≈ 0 doesn’t imply ρ is approximately separablein 1-norm (Ibinson, Linden, Winter ’08) • Good news • I(A:B|C) still defined • Chain rule, etc. still hold • I(A:B|C)ρ=0 impliesρ is separable • (Hayden, Jozsa, Petz, Winter‘03)

47. Proving the Bound Obs: I(A:B|E)≥0 is strong subadditivity inequality (Lieb, Ruskai ‘73) Chain rule: 2log|A| > I(A:B1…Bk) = I(A:B1)+I(A:B2|B1)+…+I(A:Bk|B1…Bk-1) For ρABk-extendible: Thm(B., Christandl, Yard ’11)

48. Conditional Mutual Information Bound • Coding Theory • Strong subadditivity of von Neumann • entropy as state redistribution rate (Devetak, Yard ‘06) • Large Deviation Theory • Hypothesis testing for entanglement (B., Plenio‘08) • ()

49. hSEPequivalent to Injective norm of 3-index tensors Minimum output entropy quantum channel, Optimal acceptance probability in QMA(2), …. 2->q norms of projectors: (||*||2->q for q > 2 are hypercontractive norms: useful in Markov Chain Monte Carlo (rapidly mixing condition), etc)

50. Quantum Bound on SoS Implies (Barak, B., Harrow, Kelner, Steurer, Zhou ‘12) 1. For n x n matrix A can compute with O(log(n)ε-3) rounds of SoS a number xs.t. 2. nO(ε)-level SoS solves ε-Small Set Expansion Problem (closely related to unique games). Alternative algorithm to (Arora, Barak, Steurer’10) 3.Open Problem: Improvement in bound (additive -> multiplicative error) would solve SSE in exp(O(log2(n))) time. Can formulate it as a quantum information-theoretic problem