‘comparing the Degree of Incompatibility Inherent in Probabilistic Physical Theories’

We introduce a new way of quantifying the degrees of incompatibility of two observables in a probabilistic physical theory and, based on this, a global measure of the degree of incompatibility inherent in such theories, across all observable pairs. This opens up a novel and flexible way of comparing probabilistic theories with respect to the nonclassical feature of incompatibility, raising many interesting questions, some of which will be answered here. We show that quantum theory contains observables that are as incompatible as any probabilistic physical theory can have if arbitrary pairs of observables are considered. If one adopts a more refined measure of the degree of incompatibility, for instance, by restricting the comparison to binary observables, it turns out that there are probabilistic theories whose inherent degree of incompatibility is greater than that of quantum mechanics. To be published in: Europhysics Letters (2013). Quantum theory has a number of important features not known in classical physics, ranging from the superposition and indeterminacy principles formulated by the pioneers to the more recently discovered no-cloning and no-broadcasting theorems. It is an old problem to identify operationally significant properties of quantum theory that distinguish it from other probabilistic theories. In recent years many features have been under intensive investigation from this perspective, including information processing [1], optimal state discrimination [2], entropy [3], purification [4] and discord [5]. It has been found that some properties are quite generally valid in any nonclassical (no-signaling) probabilistic theories while others are specifically quantum. The existence of pairs of incompatible observables marks one of the most striking distinctions between quantum theory and classical physical theories. There are many manifestations of incompatibility, perhaps the most famous being the Heisenberg uncertainty principle [6]. However, there are many nonclassical probabilistic theories which also possess incompatible observables, and it will be of interest to compare quantum theory with alternative theories with respect to the feature of incompatibility. To this end, we define the joint measurability region of any given pair of observables in a probabilistic theory. The joint measurability region describes the amount of added noise needed to make the observables jointly measurable. The global joint measurability feature of a probabilistic theory can then be characterized as the intersection of all the joint measurability regions associated with the theory. We demonstrate that quantum theory contains observables that are as incompatible as observables in any probabilistic theory can be. Hence, we can say that, in a global sense, quantum theory has as great a degree of incompatibility as any other probabilistic theory. But if only binary observables are considered, the degree of incompatibility inherent in quantum theory is limited and we give an example of a probabilistic theory that contains maximally incompatible binary observables. Our aim is thus to compare the incompatibility of pairs of observables in different probabilistic physical theories. We first need to set some minimal constraints. A probabilistic theory is a framework that provides a description of physical systems in terms of states and observables with the following general properties: ar X iv :1 21 0. 41 42 v4 [ qu an tph ] 1 9 Ju n 20 13