Quantum de Finetti theorems for local measurements

Transcription

1 Quantum de Finetti theorems for local measurements methods to analyze SDP hierarchies Fernando Brandão (UCL) Aram Harrow (MIT) arxiv:

2 motivation/warmup nonlinear optimization --> convex optimization S n = {x 2 R n : kxk 2 =1} max x2s n nx i,j=1 M i,j x i x j = max x2s n M,xx T = max 2D(n) hm, i ha, Bi = nx A i,j B i,j i,j=1 D(n) = conv {xx T : x S n } = density matrices

3 a harder problem max x2s n X i 1,i 2,j 1,j 2 M i1 i 2,j 1 j 2 x i1 x i2 x j1 x j2 = max x2s n M,xx T xx T (A B) i1 i 2,j 1 j 2 = A i1,j 1 B i2,j 2 = max{hm, i : 2 SepSym(n, n)} SepSym(n,n) = conv {xx T xx T : x S n } Sep(n,n) = conv {xx T yy T : x,y S n } D(n 2 )

4 polynomial optimization D(n) n = conv {xx T : x S n } SepSym(n,n) = conv {xx T xx T : x S n } EASY HARD tensor norms 2->4 norm small-set expansion need to find relaxation!

5 k-extendable relaxation want σ SepSym(n,n) = conv {xx T xx T : x S n } relax to 2 D(n k ) i1...i k,j 1...j k = i (1)...i (k),j 0 (1)...j 0 (k) π,π S k ideally i1...i k,j 1...j k = x i1 x ik x j1 x jk recover ρ D(n 2 ) i1 i 2,j 1 j 2 = X i1 i 2 i 3...i k,j 1 j 2 i 3...i k i 3,...,i k why? 1. partial trace = quantum analogue of marginal distribution 2. using i x 2 i = 1 constraint

6 why should this work? physics explanation: monogamy of entanglement only separable states are infinitely sharable math explanation: = X i (x i x T i ) k = X i,j x i x T j x i x T 2(k 2) j hx i,x j i! X i x i x T i x i x T i as kà

7 run-time = n O(k) convergence rate dist(k-extendable, SepSym(n,n)) = f(k,n) =?? trace dist(ρ,σ) = max 0 M I hm,ρ-σi ~ n/k à n O(n) time [Brandão, Christandl, Yard; STOC 11] distance ~ (log(n)/k) 1/2 for M that are 1-LOCC à time n O(log(n)) Def of 1-LOCC M = i A i B i such that 0 A i I 0 B i i B i = I Pr[accept i] = ha i B i,ρi / Pr[i] ρ i Pr[i] = hi B i,ρi

8 our results 1. simpler proof of BCY 1-LOCC bound 2. extension to multipartite states 3. dimension-independent bounds if Alice is non-adaptive 4. extension to non-signaling distributions 5. explicit rounding scheme 6. (next talk) version without symmetry applications 1. optimal algorithm for degree- n poly optimization (assuming ETH) 2. optimal algorithm for approximating value of free games 3. hardness of entangled games 4. QMA = QMA with poly(n) unentangled Merlins & 1-LOCC measurements 5. pretty good tomography without independence assumptions 6. convergence of Lasserre 7. multipartite separability testing

9 proof sketch Further restrict to LO measurements ρ M = a,b Υ ab A a B b 0 Υ ab 1 a A a = I a Pr[a,b] = h½, A a B b i b b B b = I Goal: max a,b Pr[a,b] γ a,b exact solutions (ρ SepSym): Pr[a,b] = λ i q i (a) r i (b) Pr[accept a,b] = γ ab

10 Pr[a,b] = h½, A a B b i Goal: max a,b Pr[a,b] γ a,b exact solutions (ρ SepSym): Pr[a,b] = λ i q i (a) r i (b) rounding M = a,b Υ ab A a B b 0 Υ ab 1 a A a = I b B b = I relaxation Pr[a, b] = X b 2,...,b k p a,b,b2,...,b k p a,b1,...,b k = h,a a B b1 B bk i proof idea good approximation if Pr[a,b 1 ] ε Pr[a] Pr[b 1 ] otherwise H(a b 1 ) < H(a) ε 2

11 information theory [Raghavendra-Tan, SODA 12] log(n) I(a:b 1 b k ) = I(a:b 1 ) + I(a:b 2 b 1 ) + + I(a:b k b 1 b k-1 ) I(a:b j b 1 b j-1 ) log(n)/k for some j ρ Sep for this particular measurement Note: Brandão-Christandl-Yard based on quantum version of I(a:b c).

12 open questions 1. Improve 1-LOCC to SEP would imply QMA = QMA with poly(n) Merlins and quasipolynomial-time algorithms for tensor problems 2. Better algorithms for small-set expansion / unique games 3. Make use of partial transpose symmetry 4. Understand quantum conditional mutual information 5. extension to entangled games that would yield NEXP MIP *. (see paper) 6. More counter-examples / integrality gaps.