Generalised probabilistic theories and the extension complexity of polytopes
Serge Massar
Physical Theories
• Classical• Quantum• Generalised Probablisitic
Theories (GPT)
Factorisation ofCommunication / Slack Matrix
• Linear• SDP• Conic
ExtendedFormulations
• linear• SDP• Conic
Polytopes& Combinat.Optimisation
Comm.Complexity
From Foundations to Combinatorial Optimisation
Physical Theories
• Classical• Quantum• Generalised Probablisitic
Theories (GPT)
Factorisation ofCommunication / Slack Matrix
• Linear• SDP• Conic
ExtendedFormulations
• 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. 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
Physical Theories
• Classical• Quantum• Generalised Probablisitic
Theories (GPT)
Factorisation ofCommunication / Slack Matrix
• Linear• SDP• Conic
ExtendedFormulations
• linear• SDP• Conic
Polytopes& Combinat.Optimisation
Comm.Complexity
From Foundations to Combinatorial Optimisation
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)
A bit of geometry
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
Normalisedstates
0
Classical Theory
• • 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)
Quantum Theory
• • u=I=identity matrix• Normalised states = density matrices• Measurements = POVM
Lorentz ConeSecond Order Cone Programming
• CSOCP={x = (x0, x1,…,xn) such that x12+x2
2+…+xn2≤ x0
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 include SOCP include SDP
• Status?
Completely Positive and Co-positive Cones
Open Question.
• Other interesting families of Cones ?
One way communication complexity.
Alice Bob : M(b)
a b
w(a)
r
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
Alice Bob : M
a
w(a)
r
Proof 1: Refining MeasurementsGeneralised Probabilistic Theory GPT(C,u)• States • Measurement
• Refining measurements– If ei=pfi+(1-p)gi with – then we can refine the measurement to containeffect pfi and effect (1-p)gi rather than ei
• Theorem: Measurements can be refined so that all effects are extreme points of C* (Krein-Milman theorem)
Proof 2: Extremal MeasurementsGeneralised Probabilistic Theory GPT(C,u)• States • Measurement
• Convex combinations of measurements:– M1={ei} & M2={fi} – pM1+(1-p)M2={pei+(1-p)fi}
• If has m>n outcomes– Carathéodory: Then there exists a subset of size n, such that – Hence M=pM1+(1-p)M2 & M1 has n outcomes & M2 has m-1 outcomes.
• By recurrence: all measurements can be written as convex combination of measurements with at most n effects.
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?
Alice Bob : M
a
w(a)r
Physical Theories
• Classical• Quantum• Generalised Probablisitic
Theories (GPT)
Factorisation ofCommunication / Slack Matrix
• Linear• SDP• Conic
ExtendedFormulations
• linear• SDP• Conic
Polytopes& Combinat.Optimisation
Comm.Complexity
From Foundations to Combinatorial Optimisation
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 Cab on inputs a,bIf and only ifCone factorisation of
Alice
a b
w(a)
r(i,b)≥0
1 bit {0,1}
Different Cone factorisations
Theorem: Randomised one way communication with positive outcomes using GPT(C,u) and one bit of classical communication produces on average Cab on inputs a,bIf and only ifCone factorisation of
Alice
a b
w(a)
r(i,b)≥0
Physical Theories
• Classical• Quantum• Generalised Probablisitic
Theories (GPT)
Factorisation ofCommunication / Slack Matrix
• Linear• SDP• Conic
ExtendedFormulations
• linear• SDP• Conic
Polytopes& Combinat.Optimisation
Comm.Complexity
From Foundations to Combinatorial Optimisation
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)
Polytope
• P = conv {vertices} = {x : Aex < be}
v e
Combinatorial Polytopes• Travelling Salesman Problem (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
• 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
Physical Theories
• Classical• Quantum• Generalised Probablisitic
Theories (GPT)
Factorisation ofCommunication / Slack Matrix
• Linear• SDP• Conic
ExtendedFormulations
• linear• SDP• Conic
Polytopes& Combinat.Optimisation
Comm.Complexity
From Foundations to Combinatorial Optimisation
Extended Formulations• View polytope as projection of a simpler
object in a higher dimensional space.
p
Q=extended formulation
P=polytope
Linear Extensions: the higher dimensional object is a polytope
p
Q=extended formulation
P=polytope
Size of linear extended formulation = # of facets of Q
Conic extensions: Extended object= intersection of cone and hyperplane.
p
Cone=C
Q
Polytope P
Conic extensions
p
Cone=C
• Linear extensions– positive orthant
• SDP extensions– cone of SDP matrices
• Conic extensions– C=cone in Rn
• 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
Physical Theories
• Classical• Quantum• Generalised Probablisitic
Theories (GPT)
Factorisation ofCommunication / Slack Matrix
• Linear• SDP• Conic
ExtendedFormulations
• linear• SDP• Conic
Polytopes& Combinat.Optimisation
Comm.Complexity
From Foundations to Combinatorial Optimisation
Slack Matrix of a Polytope
• P = conv {vertices} = {x : Aex – be≥0}• Slack Matrix– Sve= distance between v and e = Aexv – be
v e
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 Sev by sending GPT(C,u) states.
Alice Bob
e v
GPT(C)
s : <s>=Sev
Physical Theories
• Classical• Quantum• Generalised Probablisitic
Theories (GPT)
Factorisation ofCommunication / Slack Matrix
• Linear• SDP• Conic
ExtendedFormulations
• 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
A Classical versus Quantum gap
Alice Bob
a b
Classical/Quantum Communication
m : <m>=Mab
Theorem: Linear Extension Complexity of Correlation Polytope=
Alice Bob
a b
Classical Communication
m : <m>=Mab
Linear extension complexity of polytopes
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)
Physical Theories
• Classical• Quantum• Generalised Probablisitic
Theories (GPT)
Factorisation ofCommunication / Slack Matrix
• Linear• SDP• Conic
ExtendedFormulations
• 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: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.
Recall:Completely Positive and Co-positive Cones
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
Polynomialy definable 0/1 polytopes
Polynomialy definable 0/1-polytopes
• Theorem (Maksimenko2012): All polynomialy definable 0/1-polytopes in Rd are projections of faces of the correlation polytope COR(poly(d)).
• Corollary: All polynomialy definable 0/1-polytopes in Rd 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
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
Physical Theories
• Classical• Quantum• Generalised Probablisitic
Theories (GPT)
Factorisation ofCommunication / Slack Matrix
• Linear• SDP• Conic
ExtendedFormulations
• 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. 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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