Compatibility for probabilistic theories (Q2822856)

The existence of incompatible sets of observables is a feature that sets quantum mechanics apart from classical physical theories. Casting classical and quantum theories as instances of a general family of probabilistic theories (PT), each defined by its own \(\sigma \)-convex set \(\mathcal K\) of states living in a real Banach space \(\mathcal V\), the question has been asked how one might define the degree of incompatibility inherent in such theories. The answer could contribute to an axiomatic characterisation of quantum mechanics.NEWLINENEWLINENEWLINEAn observable on a PT is defined as an affine map from \(\mathcal K\) to the set of probability measures on a Borel \(\sigma \)-algebra \(\mathcal B (\mathbb R^m)\). Two (or more) observables are compatible if they arise as margins of a common third observable, referred to as a joint observable, defined on the appropriate product algebra. These notions of observables and their compatibility generalize the quantum mechanical concept of observables as spectral measures and their commutativity. The set of observables with a common range \(\mathcal B(\mathbb R^m)\) is convex and contains trivial elements, which are the constant affine maps from \(\mathcal K\) to some fixed probability measure and represent completely non-informative, noisy measurements. Intuitively, the degree of compatibility of (say) two observables \(M_1\), \(M_2\) can then be defined by how much noise (suitable trivial observables \(T_1,T_2\)) one has to add to them so as to make the resulting pairs \(M_1'=\lambda M_1+(1-\lambda) T_1\), \(M_1'=\lambda M_2+(1-\lambda) T_2\) compatible.NEWLINENEWLINENEWLINEThe present work continues a relatively new line of investigation into the problem of characterising observables of general PTs and classifying such theories according to their degrees of inherent (in)compatibility. The paper begins with a review of a recently proposed definition of the compatibility region of a finite ordered set observables of a PT. The compatibility region of an ordered \(n\)-tuple of observables \((M_1,M_2,\dots ,M_n)\) (all defined on the same domain) is defined as the set of \((\lambda_1,\lambda_2,\dots \lambda_n)\in [0,1]^n\) such that the observables \(M_k'=\lambda_k M_k+(1-\lambda_k)T_k\) are compatible for suitable choices of trivial observables \(T_k\).NEWLINENEWLINENEWLINEDegrees of compatibility of observables -- and ultimately of different PTs -- are compared in terms of the ``sizes'' of the associated compatibility regions. The paper proceeds to simplify this multidimensional concept by introducing a concept of compatibility intervals for a pair of observables, which gives a more easily tractable measure. This is used to define a new compatibility index whose values range between \(0\) (minimal compatibility, shown to be realized by quantum mechanics) and \(1\) (classical PTs).NEWLINENEWLINENEWLINEThe paper concludes by showing that in a PT, observables can be represented as vector-valued measures, generalizing the quantum mechanical notion of an observable as a positive-operator-valued measure.