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

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

2 Physical Theories Classical Quantum Generalised Probablisitic Theories (GPT) Factorisation of Communication / Slack Matrix Linear SDP Conic Extended Formulations linear SDP Conic Polytopes & Combinat. Optimisation Comm. Complexity From Foundations to Combinatorial Optimisation

3 Physical Theories Classical Quantum Generalised Probablisitic Theories (GPT) Factorisation of Communication / Slack Matrix Linear SDP Conic Extended Formulations linear SDP Conic Polytopes & Combinat. Optimisation Comm. Complexity From Foundations to Combinatorial Optimisation  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  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: arXiv:

4 Physical Theories Classical Quantum Generalised Probablisitic Theories (GPT) Factorisation of Communication / Slack Matrix Linear SDP Conic Extended Formulations linear SDP Conic Polytopes & Combinat. Optimisation Comm. Complexity From Foundations to Combinatorial Optimisation

5 Generalised Probabilistic Theories Minimal framework to build theories – States = convex set – Measurements: Predict probability of outcomes Adding axioms restricts to Classical or Quantum Theory – Aim: find « Natural » axioms for quantum theory. (Fuchs, Brassar, Hardy, Barrett, Masanes Muller, D’Ariano etal, etc…) GPTs with « unphysical » behavior -> rule them out. – PR boxes make Communication Complexity trivial (vanDam 05) – Correlations that violate Tsirelson bound violate Information Causality (Pawlowski et al 09)

6 A bit of geometry

7 Generalised Probabilistic Theories Mixture of states = state – State space is convex Theory predicts probability of outcome of measurement. Generalised Probabilistic Theory GPT(C,u) Space of unnormalised states = Cone Effects belong to dual Cone Normalisation – Unit – Normalised state – Measurement – Probability of outcome i : C* C. u Normalised states 0

8 Classical Theory u=(1,1,1,…,1) Normalised state  =(p 1,p 2,…,p n ) Probability distribution over possible states Canonical measurement={e i } e i =(0,..,0,1,0,..,0)

9 Quantum Theory u=I=identity matrix Normalised states = density matrices Measurements = POVM

10 Lorentz Cone Second Order Cone Programming C SOCP ={x = (x 0, x 1,…,x n ) such that x 1 2 +x 2 2 +…+x n 2 ≤ x 0 2 } Lorentz cone has a natural SDP formulation -> subcone of the cone of SDP matrices Can be arbitrarily well approximated using linear inequalities Linear programs  SOCP  SDP Status?

11 Completely Positive and Co-positive Cones

12 Open Question. Other interesting families of Cones ?

13 One way communication complexity. AliceBob : M(b) ab  (a) r

14 Classical Capacity. Holevo Theorem: – How much classical information can be stored in a GPT state? Max I(A:R) ? – At most log(n) bits can be stored in AliceBob : M a  (a) r

15 Proof 1: Refining Measurements Generalised Probabilistic Theory GPT(C,u) States Measurement Refining measurements – If e i =pf i +(1-p)g i with – then we can refine the measurement to contain effect pf i and effect (1-p)g i rather than e i Theorem: Measurements can be refined so that all effects are extreme points of C* (Krein-Milman theorem)

16 Proof 2: Extremal Measurements Generalised Probabilistic Theory GPT(C,u) States Measurement Convex combinations of measurements: – M 1 ={e i } & M 2 ={f i } – pM 1 +(1-p)M 2 ={pe i +(1-p)f i } If has m>n outcomes – Carathéodory: Then there exists a subset of size n, such that – Hence M=pM 1 +(1-p)M 2 & M 1 has n outcomes & M 2 has m-1 outcomes. By recurrence: all measurements can be written as convex combination of measurements with at most n effects.

17 Proof 3: Classical Capacity of GPT Holevo Theorem for – Refining a measurement and decomposing measurement into convex combination can only increase the capacity of the channel – Capacity of channel  log( # of measurement outcomes)  Capacity of channel ≤ log(n) bits OPEN QUESTION: – Get better bounds on the classical capacity for specific theories? AliceBob : M a  (a) r

18 Physical Theories Classical Quantum Generalised Probablisitic Theories (GPT) Factorisation of Communication / Slack Matrix Linear SDP Conic Extended Formulations linear SDP Conic Polytopes & Combinat. Optimisation Comm. Complexity From Foundations to Combinatorial Optimisation

19 Randomised one way communication complexity with positive outcomes Theorem: Randomised one way communication with positive outcomes using GPT(C,u) and one bit of classical communication produces on average C ab on inputs a,b If and only if Cone factorisation of Alice a b  (a) r(i,b)  1 bit {0,1}

20 Different Cone factorisations Theorem: Randomised one way communication with positive outcomes using GPT(C,u) and one bit of classical communication produces on average C ab on inputs a,b If and only if Cone factorisation of Alice a b  (a) r(i,b) 

21 Physical Theories Classical Quantum Generalised Probablisitic Theories (GPT) Factorisation of Communication / Slack Matrix Linear SDP Conic Extended Formulations linear SDP Conic Polytopes & Combinat. Optimisation Comm. Complexity From Foundations to Combinatorial Optimisation

22 Background: solving NP by LP? Famous P-problem: linear programming (Khachian’79) Famous NP-hard problem: traveling salesman problem A polynomial-size LP for TSP would show P = NP Swart’86–87 claimed to have found such LPs Yannakakis’88 showed that any symmetric LP for TSP needs exponential size Swart’s LPs were symmetric, so they couldn’t work 20-year open problem: what about non-symmetric LP? There are examples where non-symmetry helps a lot (Kaibel’10) Any LP for TSP needs exponential size (Fiorini et al 12)

23 Polytope P = conv {vertices} = {x : A e x < b e } v e

24 Combinatorial Polytopes Travelling Salesman Problem (TSP) polytope – R n(n-1)/2 : one coordinate per edge of graph – Cycle C : v C =(1,0,0,1,1,…,0) – P TSP =conv{v C } – Shortest cycle: min Correlation polytope – Bell polytope with 2 parties, N settings, 2 outcomes Linear optimisation over these polytopes is NP Hard Deciding if a point belongs to the polytope is NP Hard

25 Physical Theories Classical Quantum Generalised Probablisitic Theories (GPT) Factorisation of Communication / Slack Matrix Linear SDP Conic Extended Formulations linear SDP Conic Polytopes & Combinat. Optimisation Comm. Complexity From Foundations to Combinatorial Optimisation

26 Extended Formulations View polytope as projection of a simpler object in a higher dimensional space.  Q=extended formulation P=polytope

27 Linear Extensions: the higher dimensional object is a polytope  Q=extended formulation P=polytope Size of linear extended formulation = # of facets of Q

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

29 Conic extensions  Cone=C Linear extensions – positive orthant SDP extensions – cone of SDP matrices Conic extensions – C=cone in R n 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 Q Polytope P

30 Physical Theories Classical Quantum Generalised Probablisitic Theories (GPT) Factorisation of Communication / Slack Matrix Linear SDP Conic Extended Formulations linear SDP Conic Polytopes & Combinat. Optimisation Comm. Complexity From Foundations to Combinatorial Optimisation

31 Slack Matrix of a Polytope P = conv {vertices} = {x : A e x – b e  } Slack Matrix – S ve = distance between v and e = A e x v – b e v e

32 Factorisation Theorem (Yannakakis88) Theorem: Polytope P has Cone C extension Iff Slack matrix has Conic factorisation – Iff Alice and Bob can solve communication complexity problem based on S ev by sending GPT  C,u) states. AliceBob ev GPT(C) s : =S ev

33 Physical Theories Classical Quantum Generalised Probablisitic Theories (GPT) Factorisation of Communication / Slack Matrix Linear SDP Conic Extended Formulations linear SDP Conic Polytopes & Combinat. Optimisation Comm. Complexity From Foundations to Combinatorial Optimisation 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 AliceBob ab Classical/Quantum Communication m : =M ab

35 Theorem: Linear Extension Complexity of Correlation Polytope= AliceBob ab Classical Communication m : =M ab

36 Linear extension complexity of polytopes

37 OPEN QUESTION? Prove that SDP (Quantum) extension complexity of TSP, Correlation, etc.. polytopes is exponential – Strongly conjectured to be true – The converse would almost imply P=NP – Requires method to lower bound quantum communication complexity in the average output model (cannot give the parties shared randomness)

38 Physical Theories Classical Quantum Generalised Probablisitic Theories (GPT) Factorisation of Communication / Slack Matrix Linear SDP Conic Extended Formulations linear SDP Conic Polytopes & Combinat. Optimisation Comm. Complexity From Foundations to Combinatorial Optimisation S. Fiorini, S. Massar, M. K. Patra, H. R. Tiwary, Generalised probabilistic theories and the extension complexity of polytopes, arXiv: arXiv: 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 Correlation Polytope Theorem: The Correlation polytope COR(n) has a 2n+1 size extension for the Completely Positive Cone. – Sketch of proof: Consider arbitary linear optimisation over COR(n) Use Equivalence (Bürer2009) to linear optimisation over C* 2n+1 Implies COR(n)=projection of intersection of C* 2n+1 with hyperplane

41 Polynomialy definable 0/1 polytopes

42 Polynomialy definable 0/1-polytopes Theorem (Maksimenko2012): All polynomialy definable 0/1- polytopes in R d are projections of faces of the correlation polytope COR(poly(d)). Corollary: All polynomialy definable 0/1-polytopes in R d have poly(d) size extension for the Completely Positive Cone. – Generalises a large number of special cases proved before. – « Cook-Levin» like theorem for combinatorial polytopes

43 Summary Generalised Probabilistic Theories – Holevo Theorem for GPT Connection between Classical/Quantum/GPT communication complexity and Extension of Polytopes – Exponential Lower bound on linear extension complexity of COR, TSP polytopes – All 0/1 combinatorial polytopes have small extension for the Completely Positive Cone – Hence: GPT(Completely Positive Cone) allows exponential saving with respect to classical (conjectured quantum) communication. Use this to rule out the theory? (Of course many other reasons to rule out the theory using other axioms) OPEN QUESTIONS: Gaps between Classical/Quantum/GPT for – Other models of communication complexity? – Models of Computation

44 Physical Theories Classical Quantum Generalised Probablisitic Theories (GPT) Factorisation of Communication / Slack Matrix Linear SDP Conic Extended Formulations linear SDP Conic Polytopes & Combinat. Optimisation Comm. Complexity From Foundations to Combinatorial Optimisation  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  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: arXiv: 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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