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Quantum de Finetti theorems for local measurements

Quantum de Finetti theorems for local measurements. methods to analyze SDP hierarchies. Fernando Brandão (UCL) Aram Harrow (MIT). arXiv:1210.6367. motivation/warmup. nonlinear optimization --> convex optimization. D(n) = conv {xx T : x∈S n } = density matrices. a harder problem.

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Quantum de Finetti theorems for local measurements

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  1. Quantum de Finetti theorems for local measurements methods to analyzeSDP hierarchies Fernando Brandão (UCL) Aram Harrow (MIT) arXiv:1210.6367

  2. motivation/warmup nonlinear optimization --> convex optimization D(n) = conv {xxT : x∈Sn} = density matrices

  3. a harder problem SepSym(n,n) = conv {xxT ⊗ xxT : x∈Sn} ⊂ Sep(n,n) = conv {xxT ⊗ yyT : x,y∈Sn} ⊂ D(n2)

  4. polynomial optimization EASY D(n)n = conv {xxT : x∈Sn} HARD SepSym(n,n) = conv {xxT ⊗ xxT : x∈Sn} ≈ tensor norms ≈ 2->4 norm ≈ small-set expansion need to findrelaxation!

  5. k-extendable relaxation want σ∈SepSym(n,n) = conv {xxT ⊗ xxT : x∈Sn} relax to ∀π,π’∈Sk ideally recover ρ∈ D(n2) why? partial trace = quantum analogue of marginal distribution using ∑ixi2 = 1 constraint

  6. why should this work? physics explanation:“monogamy of entanglement”only separable states are infinitely sharable math explanation: as k∞

  7. convergence rate run-time = nO(k) dist(k-extendable, SepSym(n,n)) = f(k,n) = ?? trace dist(ρ,σ) = max0≤M≤IhM,ρ-σi ～ n/k  nO(n) time [Brandão, Christandl, Yard; STOC ‘11] distance ～ (log(n)/k)1/2 for M that are 1-LOCC time nO(log(n)) i ρ Def of 1-LOCCM = ∑i Ai ⊗ Bi such that0 ≤ Ai ≤ I0 ≤ Bi∑i Bi = I Pr[accept | i] =hAi⊗Bi,ρi / Pr[i] Pr[i] = hI⊗Bi,ρi

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

  9. proof sketch Further restrict to LO measurements M = ∑a,b Υab Aa ⊗ Bb ρ 0 ≤ Υab ≤ 1 ∑a Aa = I ∑b Bb = I Pr[a,b] = h½, Aa­ Bbi b a Goal: max ∑a,b Pr[a,b] γa,b exact solutions (ρ∈SepSym):Pr[a,b] = ∑λi qi(a) ri(b) Pr[accept | a,b] = γab

  10. rounding Pr[a,b] = h½, Aa­ Bbi Goal: max ∑a,b Pr[a,b] γa,b M = ∑a,b Υab Aa ⊗ Bb exact solutions (ρ∈SepSym):Pr[a,b] = ∑λi qi(a) ri(b) 0 ≤ Υab ≤ 1 ∑a Aa = I relaxation ∑b Bb = I • proof idea • good approximation if Pr[a,b1] ≈ε Pr[a] ⋅ Pr[b1] • otherwiseH(a|b1) < H(a) – ε2

  11. information theory [Raghavendra-Tan, SODA ’12] log(n) ≥ I(a:b1 … bk) = I(a:b1) + I(a:b2|b1) + … + I(a:bk|b1…bk-1) ∴ I(a:bj|b1…bj-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 Improve 1-LOCC to SEPwould imply QMA = QMA with poly(n) Merlinsand quasipolynomial-time algorithms for tensor problems Better algorithms for small-set expansion / unique games Make use of “partial transpose” symmetry Understand quantum conditional mutual information extension to entangled games that would yieldNEXP ⊆ MIP*. (see paper) More counter-examples / integrality gaps.

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