All joint von Neumann measurements on a quantum state admit a quasi-classical probability model

Abstract

We prove that the Hilbert space description of all joint von Neumann measurements on a quantum state can be reproduced in terms of a single measure space ({\Omega}, F, {\mu}) with a normalized real-valued measure {\mu}, that is, in terms of a new general probability model, the quasi-classical probability model, developed in [Loubenets: J. Math. Phys. 53 (2012), 022201; J. Phys. A: Math. Theor. 45 (2012), 185306]. In a quasi-classical probability model for all von Neumann measurements, a random variable models the corresponding quantum observable in all joint measurements and depends only on this quantum observable. This mathematical result sheds a new light on some important issues of quantum randomness discussed in the literature since the seminal article (1935) of Einstein, Podolsky and Rosen.