Excluding joint probabilities from quantum theory

Abstract

Quantum theory does not provide a unique definition for the joint probability of two noncommuting observables, which is the next important question after the Born's probability for a single observable. Instead, various definitions were suggested, e.g. via quasi-probabilities or via hidden-variable theories. After reviewing open issues of the joint probability, we relate it to quantum imprecise probabilities, which are non-contextual and are consistent with all constraints expected from a quantum probability. We study two non-commuting observables in a two-dimensional Hilbert space and show that there is no precise joint probability that applies for any quantum state and is consistent with imprecise probabilities. This contrasts to theorems by Bell and Kochen-Specker that exclude joint probabilities for more than two non-commuting observables, in Hilbert space with dimension larger than two. If measurement contexts are included into the definition, joint probabilities are not anymore excluded, but they are still constrained by imprecise probabilities.

Quantum non-locality co-exists with locality

“…[14] shows that it is also consistent with the quantum conditional (two-time) probability. We emphasize that (16) are respectively the minimal and maximal operators holding (10)(11)(12)(13)(14)(15)). If some of those conditions are omitted, the imprecise probabilities can only become more precise.…”

“…where U is a unitary operator: U U † = I. For further features of the imprecise probability see [13,14]. Ref.…”

“…usual) probability within quantum mechanics, they do have an imprecise (i.e. interval-valued) joint probability [13,14]. Then non-locality is seen in the fact that the imprecise joint probability operator calculated for tensor products of single-particle (local) observables does not reduce to a tensor product of single-particle operators.…”

“…Eqs. (14,15) also refer to the consistency [in the sense of (9)] with the precise probability, because the latter is well-defined not only for [P, Q] = 0, but also under conditions (15), where it amounts to tr(ρP Q). The latter can be calculated as an average tr(ρ P Q+QP 2 ) of the Hermitean operator P Q+QP 2 .…”

Minimizing Backaction through Entangled Measurements

“…Measurement incompatibility can also be described by constructing joint (quasi)probability distributions, with the condition that the marginal distributions be given by the Born rule [32][33][34][35][36][37]. Quasiprobability distributions, generically arising in (sequential) weak measurement scenarios [19,[38][39][40][41][42][43][44], necessarily feature negative probabilities [14,34,[45][46][47][48][49] (which is related to contextuality [50][51][52]); joint distributions that do not feature negative probabilities require going beyond the axiomatics of standard "precise" probability theory [53].…”