Generalised probabilistic theories and
the extension complexity of polytopes
Serge Massar
From Foundations to Combinatorial Optimisation
Physical Theories
Comm.
Complexity
•
•
•
Classical
Quantum
Generalised Probablisitic
Theories (GPT)
Factorisation of
Communication / Slack
Matrix
•
•
•
Linear
SDP
Conic
Extended
Formulations
•
•
•
linear
SDP
Conic
Polytopes
& Combinat.
Optimisation
From Foundations to Combinatorial Optimisation
Physical Theories
Comm.
Complexity
Factorisation of
Communication / Slack
Matrix
Classical
Quantum
Generalised Probablisitic
Theories (GPT)



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
Linear
SDP
Conic
•
•
•
Polytopes
& Combinat.
Optimisation
•
•
•

•
•
•
Extended
Formulations
linear
SDP
Conic
From Foundations to Combinatorial Optimisation
Physical Theories
Comm.
Complexity
•
•
•
Classical
Quantum
Generalised Probablisitic
Theories (GPT)
Factorisation of
Communication / Slack
Matrix
•
•
•
Linear
SDP
Conic
Extended
Formulations
•
•
•
linear
SDP
Conic
Polytopes
& Combinat.
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
Cone C Í R n
x,x' ÎC & p,p' ³ 0 then px+p'x' ÎC
Dual Cone C* Í R n
C*= { y : x, y ³ 0 for all x ÎC}
Examples:
C=C*= { x ÎR n : xi ³ 0} = positive orthant
C=C*= { x ÎR n´n : x ³ 0} = cone of SDP matrices
But in general C ¹ C*
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
C
w ÎC
Normalised
states
• Effects belong to dual Cone
e ÎC *
• Normalisation
–
–
–
–
Unit u ÎC *
0
Normalised state w ,u = 1
Measurement M = ìíe ÎC* : å e = u üý
î
þ
Probability of outcome
i : P(i | w
) = ei ,w
i
i
i
.u
C*
Classical Theory
• C=C*={ x ÎR : xi ³ 0} = positive orthant
• u=(1,1,1,…,1)
n
• Normalised state w=(p1,p2,…,pn)
Probability distribution over possible states
• Canonical measurement={ei}
ei=(0,..,0,1,0,..,0)
P(i | w) = ei ,w = pi
Quantum Theory
•
•
•
•
C=C*= { x ÎRn´n : x ³ 0} = cone of SDP matrices
u=I=identity matrix
Normalised states = density matrices
Measurements = POVM
Lorentz Cone
Second Order Cone Programming
• CSOCP={x = (x0, x1,…,xn) such that x12+x22+…+xn2≤ x02}
• 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
Co-positive cone
Cn = { X ÎR n´n : yT Xy ³ 0, y ÎR+n }
Completely positive cone
ì
ü
Cn* = í X ÎR n´n : X = å ykT yk , yk ÎR+n ý
k
î
þ
=cone of quantum states with real positive coefficients
Remarks:
-linear programming over C*n or Cn is NP hard
-deciding if x ÎC*n (or C n ) is NP hard
Status?
Open Question.
• Other interesting families of Cones ?
One way communication complexity.
a
b
Alice
w(a)
Bob : M(b)
r
Classical Capacity.
a
Alice
w(a)
Bob : M
r
• 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 w ÎC Í Rn
Proof 1: Refining Measurements
Generalised Probabilistic Theory GPT(C,u)
• States w ÎC Í Rn
• Measurement M = ìíe ÎC* Í R : å e = u üý
n
î
i
i
i
þ
• Refining measurements
– If ei=pfi+(1-p)gi with fi , gi ÎC * Í Rn
– then we can refine the measurement to contain
effect 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 Measurements
Generalised Probabilistic Theory GPT(C,u)
n
• States w ÎC Í R
• Measurement M = ìíei ÎC* Í Rn : å ei = u üý
î
i
þ
• Convex combinations of measurements:
– M1={ei} & M2={fi}
– pM1+(1-p)M2={pei+(1-p)fi}
•
ì
ü
n
M
=
e
ÎC*
Í
R
:
e
=
u
If
í i
å i ý has m>n outcomes
î
þ
– Carathéodory: i
n
Then there exists a subset of size n, such that å pk ek = u , pk ³ 0
k=1
– 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
a
Alice
w(a)
Bob : M
r
• Holevo Theorem for GPT (C Í Rn ,u)
– Refining a measurement and decomposing measurement into convex
combination can only increase the capacity of the channel
– Capacity of channel d log( # of measurement outcomes)
 Capacity of channel ≤ log(n) bits
• OPEN QUESTION:
– Get better bounds on the classical capacity for specific theories?
From Foundations to Combinatorial Optimisation
Physical Theories
Comm.
Complexity
•
•
•
Classical
Quantum
Generalised Probablisitic
Theories (GPT)
Factorisation of
Communication / Slack
Matrix
•
•
•
Linear
SDP
Conic
Extended
Formulations
•
•
•
linear
SDP
Conic
Polytopes
& Combinat.
Optimisation
Randomised one way communication
complexity with positive outcomes
b
a
Alice
w(a)
1 bit {0,1}
ì
ü
Bob: M (b) = íei (b) ÎC * : å ei (b) = u ý
i
î
þ
Cab = E[r | ab] = å r(i,b) w(a),ei (b) = w(a), f (b)
r(i,b)e 
i
w(a) ÎC
f (b) = å r(i,b)ei (b) ÎC *
i
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,b
If and only if
Cone factorisation of Cab = w(a), f (b) , w(a) ÎC, f (b) ÎC *
Different Cone factorisations
b
a
w(a)
Alice
ì
ü
Bob: M (b) = íei (b) ÎC * : å ei (b) = u ý
i
î
þ
r(i,b)e 
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,b
If and only if
Cone factorisation of Cab = w(a), f (b) , w(a) ÎC, f (b) ÎC *
Classical Theory: Cab = å wi (a) fi (b) with wi (a) fi (b) ³ 0 : positive rank of Cab
i
Quantum Theory: Cab = Tr w(a) f (b) with w(a), f (b) SDP matrices : SDP rank of Cab
Other Theories: Cone factorisation of Cab
From Foundations to Combinatorial Optimisation
Physical Theories
Comm.
Complexity
•
•
•
Classical
Quantum
Generalised Probablisitic
Theories (GPT)
Factorisation of
Communication / Slack
Matrix
•
•
•
Linear
SDP
Conic
Extended
Formulations
•
•
•
linear
SDP
Conic
Polytopes
& Combinat.
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 L ix with x ÎPTSP
• Correlation polytope
–
{
COR(n)=conv bT b :b Î{0,1}
n
}
• 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
From Foundations to Combinatorial Optimisation
Physical Theories
Comm.
Complexity
•
•
•
Classical
Quantum
Generalised Probablisitic
Theories (GPT)
Factorisation of
Communication / Slack
Matrix
•
•
•
Linear
SDP
Conic
Extended
Formulations
•
•
•
linear
SDP
Conic
Polytopes
& Combinat.
Optimisation
Extended Formulations
• View polytope as projection of a simpler
object in a higher dimensional space.
Q=extended formulation
p
P=polytope
Linear Extensions: the higher
dimensional object is a polytope
Q=extended formulation
p
P=polytope
Size of linear extended formulation = # of facets of Q
Conic extensions:
Extended object= intersection of cone and hyperplane.
Q
Cone=C
Precise formulation
p
Q= {(x, y) ÎR d+n : Ex + Fy = g, y ÎC }
Px (Q) = { x ÎR d : $y ÎR n & (x, y) ÎQ}
Polytope P
Conic extensions
Q
• Linear extensions
– positive orthant
C = { x ÎRn : xi ³ 0}
Cone=C
• SDP extensions
– cone of SDP matrices
C = { x ÎRn´n : x ³ 0}
p
• Conic extensions
– C=cone in Rn
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
From Foundations to Combinatorial Optimisation
Physical Theories
Comm.
Complexity
•
•
•
Classical
Quantum
Generalised Probablisitic
Theories (GPT)
Factorisation of
Communication / Slack
Matrix
•
•
•
Linear
SDP
Conic
Extended
Formulations
•
•
•
linear
SDP
Conic
Polytopes
& Combinat.
Optimisation
Slack Matrix of a Polytope
• P = conv {vertices} = {x : Aex – bee }
• 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
– Sve = Te ,Uv with Te ÎC &Uv ÎC *
• Iff Alice and Bob can solve communication
complexity problem based on Sev by sending
GPT(C,u) states.
e
v
Alice
GPT(C)
Bob
s : <s>=Sev
From Foundations to Combinatorial Optimisation
Physical Theories
Comm.
Complexity
•
•
•
Classical
Quantum
Generalised Probablisitic
Theories (GPT)
Factorisation of
Communication / Slack
Matrix
•
•
•
Linear
SDP
Conic
Extended
Formulations
•
•
•
linear
SDP
Conic
Polytopes
& Combinat.
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
a
b
Classical/Quantum Communication
Alice
a,b Î{0,1}
Bob
m : <m>=Mab
n
M ab = (1- aT b )
2
Supp(M ab ) = 0 if M ab = 0 (if a&b have one common 1 entry)
Supp(M ab ) =1 if M ab > 0 (if a&b dont have exactly one common 1 entry)
Simpler problem: # of classical bits to output Supp(M ab )
= W(n) (Razborov92)
# of classical bits required to produce on average M ab = W(n)
# of qubits to output Supp(M ab ) = O((log(n)) (deWolf03)
Theorem: Linear Extension Complexity
W(n)
of Correlation Polytope= 2
a
b
Classical Communication
Alice
a,b Î{0,1}
Bob
m : <m>=Mab
n
M ab = (1- aT b )
2
# of classical bits required to produce on average M ab = W(n)
Correlation polytope
{
COR(n)=conv bT b :b Î{0,1}
n
}
Observation: M ab = part of the slack matrix of COR(n)
Hence outputing M ab is at least as hard as outputing Slack Matrix of COR(n)
Theorem: Linear Extension Complexity of COR(n)=2W(n)
Linear extension complexity of
polytopes
Linear Extension Complexity of COR(n)=2W(n)
® Linear Extension Complexity of TSP(n)=2
W(n1/4 )
(since improved to 2 W(n) )
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)
From Foundations to Combinatorial Optimisation
Physical Theories
Comm.
Complexity
•
•
•
Classical
Quantum
Generalised Probablisitic
Theories (GPT)
Factorisation of
Communication / Slack
Matrix
•
•
•
Linear
SDP
Conic
Extended
Formulations
•
•
•
linear
SDP
Conic
Polytopes
& Combinat.
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
Co-positive cone
Cn = { X ÎR n´n : yT Xy ³ 0, y ÎR+n }
Completely positive cone
ì
ü
Cn* = í X ÎR n´n : X = å ykT yk , yk ÎR+n ý
k
î
þ
=cone of quantum states with real positive coefficients
Remarks:
-linear programmin over C*n or Cn is NP hard
-deciding if x ÎC*n (or C n ) is NP hard
Completely Positive extention of
Correlation Polytope
•
{
COR(n)=conv bT b :b Î{0,1}
n
}
• 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
0/1 polytope in R = conv{y Î{0,1} }
d
d
0/1polytope is polynomialy definable:
–Iff $ poly size circuit that, given input y Î{0,1}
d
will output 1 if y ÎP, 0 if y ÏP
Most polytopes of interest belong
to families of polynomially definable 0/1-polytopes:
COR ( n ) –TSP ( n ) –Etc…
Polynomialy definable 0/1-polytopes
0/1 polytope in R d = conv{y Î{0,1} }
d
0/1polytope is polynomialy definable:
–Iff $ poly size circuit that, given input y Î{0,1}
d
will output 1 if y ÎP, 0 if y ÏP
• Theorem (Maksimenko2012): All polynomialy definable 0/1polytopes 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
From Foundations to Combinatorial Optimisation
Physical Theories
Comm.
Complexity
Factorisation of
Communication / Slack
Matrix
•
•
•
Extended
Formulations
•
•
•
Classical
Quantum
Generalised Probablisitic
Theories (GPT)

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
Linear
SDP
Conic
•
•
•
Polytopes
& Combinat.
Optimisation
linear
SDP
Conic