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Generalised probabilistic theories and the extension complexity of polytopes

Generalised probabilistic theories and the extension complexity of polytopes. Serge Massar. From Foundations to Combinatorial Optimisation. Physical Theories Classical Quantum Generalised Probablisitic Theories (GPT). Factorisation of Communication / Slack Matrix Linear SDP Conic.

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Generalised probabilistic theories and the extension complexity of polytopes

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  1. Generalised probabilistic theories and the extension complexity of polytopes Serge Massar

  2. FromFoundations to Combinatorial Optimisation • PhysicalTheories • Classical • Quantum • GeneralisedProbablisitic • Theories (GPT) • Factorisation of • Communication / Slack • Matrix • Linear • SDP • Conic • Extended • Formulations • linear • SDP • Conic Polytopes & Combinat. Optimisation Comm. Complexity

  3. FromFoundations to Combinatorial Optimisation • PhysicalTheories • Classical • Quantum • GeneralisedProbablisitic • Theories (GPT) • Factorisation of • Communication / Slack • Matrix • Linear • SDP • Conic • Extended • Formulations • linear • SDP • Conic Polytopes & Combinat. Optimisation Comm. Complexity • M. Yannakakis, Expressing Combinatorial Problems by Linear Programs, STOC 1988 • S. Gouveia, P. Parillo, R. Rekha, Lifts of Convex Sets and Conic Factorisations, Math. Op. Res. 2013 • S. Fiorini, S. Massar, S. Pokutta, H. R. Tiwary, R. de Wolf, Linear vs. Semi definite Extended Formulations: Exponential Separation and Strong Lower Bounds, STOC 2012 • S. Fiorini, S. Massar, M. K. Patra, H. R. Tiwary, Generalised probabilistic theories and the extension complexity of polytopes, arXiv:1310.4125

  4. FromFoundations to Combinatorial Optimisation • PhysicalTheories • Classical • Quantum • GeneralisedProbablisitic • Theories (GPT) • Factorisation of • Communication / Slack • Matrix • Linear • SDP • Conic • Extended • Formulations • linear • SDP • Conic Polytopes & Combinat. Optimisation Comm. Complexity

  5. GeneralisedProbabilisticTheories • Minimal framework to buildtheories • States = convex set • Measurements: Predictprobability of outcomes • Addingaxiomsrestricts to Classical or Quantum Theory • Aim: find « Natural » axioms for quantum theory. (Fuchs, Brassar, Hardy, Barrett, Masanes Muller, D’Arianoetal, etc…) • GPTswith « unphysical » behavior-> rulethem out. • PR boxes make Communication Complexity trivial (vanDam 05) • CorrelationsthatviolateTsirelsonboundviolate Information Causality (Pawlowski et al 09)

  6. A bit of geometry

  7. GeneralisedProbabilisticTheories • Mixture of states = state • State spaceisconvex • Theorypredictsprobability of outcome of measurement. GeneralisedProbabilisticTheory GPT(C,u) • Space of unnormalised states = Cone • Effectsbelong to dual Cone • Normalisation • Unit • Normalised state • Measurement • Probability of outcomei : C Normalised states . u 0 C*

  8. ClassicalTheory • u=(1,1,1,…,1) • Normalised state w=(p1,p2,…,pn) Probability distribution over possible states • Canonical measurement={ei} ei=(0,..,0,1,0,..,0)

  9. Quantum Theory • u=I=identitymatrix • Normalisedstates = densitymatrices • Measurements= POVM

  10. Lorentz ConeSecond OrderConeProgramming • CSOCP={x = (x0, x1,…,xn) suchthat x12+x22+…+xn2≤ x02} • Lorentz cone has a natural SDP formulation -> subcone of the cone of SDP matrices • Can bearbitrarilywellapproximatedusinglinearinequalities • Linear programs include SOCP include SDP • Status?

  11. Completely Positive and Co-positive Cones

  12. Open Question. • Otherinterestingfamilies of Cones ?

  13. One way communication complexity. a b w(a) Alice Bob : M(b) r

  14. ClassicalCapacity. a • HolevoTheorem: • How muchclassical information canbestored in a GPT state? Max I(A:R) ? • Atmost log(n) bits canbestored in w(a) Alice Bob : M r

  15. Proof 1: RefiningMeasurements GeneralisedProbabilisticTheory GPT(C,u) • States • Measurement • Refiningmeasurements • If ei=pfi+(1-p)giwith • thenwecanrefine the measurement to contain effectpfi and effect (1-p)giratherthanei • Theorem: Measurementscanberefinedsothat all effects are extreme points of C* (Krein-Milmantheorem)

  16. Proof 2: ExtremalMeasurements GeneralisedProbabilisticTheory GPT(C,u) • States • Measurement • Convexcombinations of measurements: • M1={ei} & M2={fi} • pM1+(1-p)M2={pei+(1-p)fi} • If has m>n outcomes • Carathéodory: Thenthereexists a subset of size n, suchthat • Hence M=pM1+(1-p)M2 & M1 has n outcomes & M2 has m-1 outcomes. • By recurrence: all measurementscanbewritten as convexcombination of measurementswithatmost n effects.

  17. Proof 3: ClassicalCapacity of GPT a w(a) • HolevoTheorem for • Refininga measurement and decomposingmeasurementintoconvexcombinationcanonlyincrease the capacity of the channel • Capacity of channel≤ log( # of measurementoutcomes) Capacity of channel ≤log(n) bits • OPEN QUESTION: • Getbetterbounds on the classicalcapacity for specifictheories? Alice Bob : M r

  18. FromFoundations to Combinatorial Optimisation • PhysicalTheories • Classical • Quantum • GeneralisedProbablisitic • Theories (GPT) • Factorisation of • Communication / Slack • Matrix • Linear • SDP • Conic • Extended • Formulations • linear • SDP • Conic Polytopes & Combinat. Optimisation Comm. Complexity

  19. Randomised one way communication complexitywith positive outcomes b a Theorem: Randomised one way communication with positive outcomesusing GPT(C,u) and one bit of classical communication produces on average Cab on inputs a,b If and only if Cone factorisation of w(a) Alice 1 bit {0,1} r(i,b)≥0

  20. DifferentCone factorisations b a Theorem: Randomised one way communication with positive outcomesusing GPT(C,u) and one bit of classical communication produces on average Cab on inputs a,b If and only if Cone factorisation of w(a) Alice r(i,b)≥0

  21. FromFoundations to Combinatorial Optimisation • PhysicalTheories • Classical • Quantum • GeneralisedProbablisitic • Theories (GPT) • Factorisation of • Communication / Slack • Matrix • Linear • SDP • Conic • Extended • Formulations • linear • SDP • Conic Polytopes & Combinat. Optimisation Comm. Complexity

  22. Background: solving NP by LP? • FamousP-problem: linearprogramming (Khachian’79) • Famous NP-hard problem: traveling salesmanproblem • A polynomial-size LP for TSP would show P = NP • Swart’86–87 claimed to have foundsuchLPs • Yannakakis’88 showedthatanysymmetric LP for TSP needsexponential size • Swart’sLPsweresymmetric, sotheycouldn’twork • 20-year open problem: what about non-symmetric LP? • There are exampleswhere non-symmetryhelps a lot(Kaibel’10) • AnyLP for TSP needsexponentialsize (Fiorini et al 12)

  23. Polytope • P = conv {vertices} = {x : Aex < be} v e

  24. CombinatorialPolytopes • Travelling SalesmanProblem (TSP)polytope • Rn(n-1)/2 : one coordinate per edge of graph • Cycle C : vC=(1,0,0,1,1,…,0) • PTSP=conv{vC} • Shortest cycle: min • Correlationpolytope • Bell polytopewith 2 parties, N settings, 2 outcomes • Linear optimisation over thesepolytopesis NP Hard • Deciding if a point belongs to the polytopeis NP Hard

  25. FromFoundations to Combinatorial Optimisation • PhysicalTheories • Classical • Quantum • GeneralisedProbablisitic • Theories (GPT) • Factorisation of • Communication / Slack • Matrix • Linear • SDP • Conic • Extended • Formulations • linear • SDP • Conic Polytopes & Combinat. Optimisation Comm. Complexity

  26. Extended Formulations • Viewpolytope as projection of a simplerobject in a higherdimensionalspace. Q=extended formulation p P=polytope

  27. Linear Extensions: the higherdimensionalobjectis a polytope Q=extended formulation p P=polytope Size of linearextended formulation = # of facets of Q

  28. Conic extensions: Extended object= intersection of cone and hyperplane. Q Cone=C p Polytope P

  29. Conic extensions Q • Linear extensions • positive orthant • SDP extensions • cone of SDP matrices • Conic extensions • C=cone in Rn Cone=C p Polytope P • Why this construction? • Small extensions exist for many problems • Algorithmics: optimise over small extended formulation is efficient for linear and SDP extension • Possible to obtain Lower bound on size of extension

  30. FromFoundations to Combinatorial Optimisation • PhysicalTheories • Classical • Quantum • GeneralisedProbablisitic • Theories (GPT) • Factorisation of • Communication / Slack • Matrix • Linear • SDP • Conic • Extended • Formulations • linear • SDP • Conic Polytopes & Combinat. Optimisation Comm. Complexity

  31. Slack Matrix of a Polytope • P = conv {vertices} = {x : Aex – be≥0} • Slack Matrix • Sve= distance between v and e = Aexv– be v e

  32. Factorisation Theorem(Yannakakis88) Theorem:PolytopeP has Cone C extension • IffSlack matrix has Conic factorisation • Iff Alice and Bob cansolve communication complexityproblembased on Sev by sending GPT(C,u) states. e v GPT(C) Alice Bob s : <s>=Sev

  33. FromFoundations to Combinatorial Optimisation • PhysicalTheories • Classical • Quantum • GeneralisedProbablisitic • Theories (GPT) • Factorisation of • Communication / Slack • Matrix • Linear • SDP • Conic • Extended • Formulations • linear • SDP • Conic Polytopes & Combinat. Optimisation Comm. Complexity • S. Fiorini, S. Massar, S. Pokutta, H. R. Tiwary, R. de Wolf, Linear vs. Semi definite Extended Formulations: Exponential Separation and Strong Lower Bounds, STOC 2012 • There do not exist polynomial size linear extensions of the TSP polytope

  34. A Classical versus Quantum gap a b Classical/Quantum Communication Alice Bob m : <m>=Mab

  35. Theorem: Linear Extension Complexity of CorrelationPolytope= a b Classical Communication Alice Bob m : <m>=Mab

  36. Linear extension complexity of polytopes

  37. OPEN QUESTION? • ProvethatSDP (Quantum) extension complexity of TSP, Correlation, etc.. polytopesisexponential • Stronglyconjectured to betrue • The converse wouldalmostimply P=NP • Requiresmethod to lowerbound quantum communication complexity in the average output model (cannotgive the parties sharedrandomness)

  38. FromFoundations to Combinatorial Optimisation • PhysicalTheories • Classical • Quantum • GeneralisedProbablisitic • Theories (GPT) • Factorisation of • Communication / Slack • Matrix • Linear • SDP • Conic • Extended • Formulations • linear • SDP • Conic Polytopes & Combinat. Optimisation Comm. Complexity • S. Fiorini, S. Massar, M. K. Patra, H. R. Tiwary, Generalised probabilistic theories and the extension complexity of polytopes, arXiv:1310.4125 • GPT based on cone of completely positive matrices allow exponential saving with respect to classical (conjectured quantum) communication • All combinatorial polytopes(vertices computable with poly size circuit) have poly size completely positive extension.

  39. Recall:Completely Positive and Co-positive Cones

  40. Completely Positive extention of CorrelationPolytope • Theorem: The Correlationpolytope COR(n) has a 2n+1 size extension for the Completely Positive Cone. • Sketch of proof: • Considerarbitarylinear optimisation over COR(n) • Use Equivalence (Bürer2009) to linear optimisation over C*2n+1 • ImpliesCOR(n)=projection of intersection of C*2n+1 withhyperplane

  41. Polynomialydefinable 0/1 polytopes

  42. Polynomialydefinable 0/1-polytopes • Theorem(Maksimenko2012): All polynomialydefinable 0/1-polytopes in Rdare projections of faces of the correlationpolytope COR(poly(d)). • Corollary: All polynomialydefinable 0/1-polytopes in Rd have poly(d) size extension for the CompletelyPositive Cone. • Generalises a large number of special cases provedbefore. • « Cook-Levin» liketheorem for combinatorialpolytopes

  43. Summary • GeneralisedProbabilisticTheories • HolevoTheorem for GPT • ConnectionbetweenClassical/Quantum/GPT communication complexity and Extension of Polytopes • ExponentialLowerbound on linear extension complexity of COR, TSP polytopes • All 0/1 combinatorialpolytopeshave small extension for the Completely Positive Cone • Hence: GPT(Completely Positive Cone) allowsexponentialsavingwith respect to classical (conjectured quantum) communication. • Use this to rule out the theory? (Of course manyotherreasons to rule out the theoryusingotheraxioms) • OPEN QUESTIONS: Gaps betweenClassical/Quantum/GPT for • Othermodels of communication complexity? • Models of Computation

  44. FromFoundations to Combinatorial Optimisation • PhysicalTheories • Classical • Quantum • GeneralisedProbablisitic • Theories (GPT) • Factorisation of • Communication / Slack • Matrix • Linear • SDP • Conic • Extended • Formulations • linear • SDP • Conic Polytopes & Combinat. Optimisation Comm. Complexity • M. Yannakakis, Expressing Combinatorial Problems by Linear Programs, STOC 1988 • S. Gouveia, P. Parillo, R. Rekha, Lifts of Convex Sets and Conic Factorisations, Math. Op. Res. 2013 • S. Fiorini, S. Massar, S. Pokutta, H. R. Tiwary, R. de Wolf, Linear vs. Semi definite Extended Formulations: Exponential Separation and Strong Lower Bounds, STOC 2012 • There do not exist polynomial size linear extensions of the TSP polytope • S. Fiorini, S. Massar, M. K. Patra, H. R. Tiwary, Generalised probabilistic theories and the extension complexity of polytopes, arXiv:1310.4125 • All combinatorial polytopes (vertices computable with poly size circuit) have poly size completely positive extension. • GPT based on cone of completely positive matrices allow exponential saving with respect to classical (conjectured quantum) communication

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