Entropy and information causality in general probabilistic theories

Abstract

We investigate the concept of entropy in probabilistic theories more general than quantum mechanics, with particular reference to the notion of information causality recently proposed by Pawlowski et al. (arXiv:0905.2992). We consider two entropic quantities, which we term measurement and mixing entropy. In the context of classical and quantum theory, these coincide, being given by the Shannon and von Neumann entropies respectively; in general, however, they are very different. In particular, while measurement entropy is easily seen to be concave, mixing entropy need not be. In fact, as we show, mixing entropy is not concave whenever the state space is a non-simplicial polytope. Thus, the condition that measurement and mixing entropies coincide is a strong constraint on possible theories. We call theories with this property monoentropic.Measurement entropy is subadditive, but not in general strongly subadditive. Equivalently, if we define the mutual information between two systems A and B by the usual formula I(A :where H denotes the measurement entropy and AB is a non-signaling composite of A and B, then it can happen that I(A : BC) < I (A : B). This is relevant to information causality in the sense of Pawlowski et al.: we show that any monoentropic non-signaling theory in which measurement entropy is strongly subadditive, and also satisfies a version of the Holevo bound, is informationally causal, and on the other hand we observe that Popescu-Rohrlich boxes, which violate information causality, also violate strong subadditivity. We also explore the interplay between measurement and mixing entropy and various natural conditions on theories that arise in quantum axiomatics.

Measurement interpretation and information measures in general probabilistic theory

“…They also studied the distinguishability measures and entropies in GPT [9,10]. In [11,12], the authors introduced the entropy and discussed some other problems in GPT from the perspective of information causality. The incompatibility of the two postulates, that is, the dynamical evolution postulate and the wave-packet collapse postulate, has always been a fundamental problem in the foundations of quantum mechanics.…”

Generalizing Entanglement via Informational Invariance for Arbitrary Statistical Theories

“…Working from this peculiar perspective, it can be shown that many phenomena usually considered as typically quantal, like quantum no-cloning and no-broadcasting theorems, the trade-off between state disturbance and measurement, and the existence and basic properties of entangled states, are in fact generic features of nonclassical probabilistic theories that verify a basic non-signaling constraint [1]. This is the point of departure of our present considerations.…”

“…The answer is given in, for example, [11] and [12]. Several possible applications were envisaged in [13][14][15][16][17][18][19][20].…”

Generalised Compositional Theories and Diagrammatic Reasoning

“…It is clear from the diagrammatic notation that given scalars s and s we have s • s = s ⊗ s = s ⊗ s = s • s; i.e. the scalars form a commutative monoid 3 .…”

“…The latter have recently received much attention because one can better understand a theoryquantum theory in particular-by studying it as merely a member of a broader class of theories. Notably, the study of non-locality within this framework has provided important new insights [2,3]. Whereas generalised probabilistic theories discard everything except the convex probabilistic structure, in contrast, GCTs focus on composition.…”