Generalized probabilistic theories and conic extensions of polytopes

Abstract

Generalized probabilistic theories (GPT) provide a general framework that includes classical and quantum theories. It is described by a cone C and its dual C * . We show that whether some one-way communication complexity problems can be solved within a GPT is equivalent to the recently introduced cone factorisation of the corresponding communication matrix M . We also prove an analogue of Holevo's theorem: when the cone C is contained in R n , the classical capacity of the channel realised by sending GPT states and measuring them is bounded by log n.Polytopes and optimising functions over polytopes arise in many areas of discrete mathematics. A conic extension of a polytope is the intersection of a cone C with an affine subspace whose projection onto the original space yields the desired polytope. Extensions of polytopes can sometimes be much simpler geometric objects than the polytope itself. The existence of a conic extension of a polytope is equivalent to that of a cone factorisation of the slack matrix of the polytope, on the same cone.We show that all 0/1 polytopes whose vertices can be recognized by a polynomial size circuit, which includes as a special case the travelling salesman polytope and many other polytopes from combinatorial optimisation, have small conic extension complexity when the cone is the completely positive cone.Using recent exponential lower bounds on the linear extension complexity of polytopes, this provides an exponential gap between the communication complexity of GPT based on the completely positive cone and classical communication complexity, and a conjectured exponential gap with quantum communication complexity.Our work thus relates the communication complexity of generalisations of quantum theory to questions of mainstream interest in the area of combinatorial optimisation.

Hyperdense coding and superadditivity of classical capacities in hypersphere theories

“…The upper bound of N 2 (the left-hand side of (27)) follows from the fact that the classical capacity of any GPT is bounded by the log 2 of the dimension of the state space [28], which in the case of entangled states for a pair of HST systems, as defined above, is N 2 .Thus, the gap we exhibit between single system classical capacity and two system classical capacity is close to optimal.…”

Information and communication in polygon theories

“…the collection of states, probabilities, and effects that are used. The argument is related to that used in [42] to show that extremal measurements have at most 3 effects. To prove it, first note that if ω(x) is not extremal, we can decompose it into extremal states ω(x) = i p i|x ω i .…”

“…However in [42] it was shown that by refining a measurement, and decomposing a measurement into a convex combination of other measurements, one can restrict to measurements with at most 3 effects all proportional to the extremal effects.…”

“…However this approach can be applied to all GPTs, and to simpler problems such as one way communication of a GPT state. Prior works in this direction include [42] in which it was shown that one way communication using a GPT based on the completely positive cone requires exponential classical (and conjectured quantum) communication to simulate, and [43] in which one way communication of "hypercube bits" is discussed.…”

How to make unforgeable money in generalised probabilistic theories

“…For this reason, cone programming is a natural tool to have handy when studying post-quantum theories. Although cone programming is a well-studied area of optimisation theory, it has only had a small number of applications in quantum theory [3,16,22,26,32] and in GPTs [2,14,19,21,31]. We hope this work will inspire future applications of cone programming in the study of GPTs and solidify it as an indispensable mathematical tool.…”