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arXiv:quant-ph/0304140

Quantum Joint Distributions

Todd A. Oliynyk

Published Apr 22, 2003• 0 citations
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Abstract

In this paper we provide a method for constructing joint distributions for an arbitrary set of observables on finite dimensional Hilbert spaces irrespective of whether the observables commute or not. These distributions have a number of desirable properties: they agree with the standard quantum mechanical ones if the observables commute, they also depend continuously on the observables, and under unitary transformations they behave in a reasonable manner.

DOI
10.48550/arxiv.quant-ph/0304140

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In this paper we provide a method for constructing joint distributions for an arbitrary set of observables on finite dimensional Hilbert spaces irrespective of whether the observables commute or not. These distributions have a number of desirable properties: they agree with the standard quantum mechanical ones if the observables commute, they also depend continuously on the observables, and under unitary transformations they behave in a reasonable manner.
This paper i a continuation of the first part under the same title. The author studies a joint distribution in σ\sigmaσ-finite measures for noncompatible observables of a quantum logic defined on some system of σ\sigmaσ-independent Boolean sub-σ\sigmaσ-algebras of a Boolean σ\sigmaσ-algebra. We present some necessary and sufficient conditions fot the existence of a joint distribution. In particular, it is shown that an arbitrary system of obsevables has a joint distribution in a measure iff it may be embedded into a system of compatible observables of some quantum logic. The methods used are different from those developed for finite measures. Finally, the author deals with the connection between the existence of a joint distribution and the existence of a commutator of observables, and the quantum logic of a nonseparable Hilbert space is mentioned.
A standard result in quantum mechanics is this: if two observables are commuting then they have a classical joint distribution in every state. A converse is demonstrated here: If a classical joint distribution for the pair agrees with standard quantum facts, then the observables must commute. This has consequences for some historical and recent quantum nonlocal models: they are analytically disallowed without the need for experiment, as they imply that all local observables must commute among themselves.
The situation of two independent observers conducting measurements on a joint quantum system is usually modelled using a Hilbert space of tensor product form, each factor associated to one observer. Correspondingly, the operators describing the observables are then acting non-trivially only on one of the tensor factors. However, the same situation can also be modelled by just using one joint Hilbert space, and requiring that all operators associated to different observers commute, i.e. are jointly measurable without causing disturbance. The problem of Tsirelson is now to decide the question whether all quantum correlation functions between two independent observers derived from commuting observables can also be expressed using observables defined on a Hilbert space of tensor product form. Tsirelson showed already that the distinction is irrelevant in the case that the ambient Hilbert space is of finite dimension. We show here that the problem is equivalent to the question whether all quantum correlation functions can be approximated by correlation function derived from finite-dimensional systems. We also discuss some physical examples which fulfill this requirement.
We study a generalization of the Wigner function to arbitrary tuples of hermitian operators. We show that for any collection of hermitian operators A1...An , and any quantum state there is a unique joint distribution on R^n, with the property that the marginals of all linear combinations of the operators coincide with their quantum counterpart. In other words, we consider the inverse Radon transform of the exact quantum probability distributions of all linear combinations. We call it the Wigner distribution, because for position and momentum this property defines the standard Wigner function. We discuss the application to finite dimensional systems, establish many basic properties and illustrate these by examples. The properties include the support, the location of singularities, positivity, the behavior under symmetry groups, and informational completeness.
We study a generalization of the Wigner function to arbitrary tuples of Hermitian operators. We show that for any collection of Hermitian operators A1, …, An and any quantum state, there is a unique joint distribution on Rn with the property that the marginals of all linear combinations of the Ak coincide with their quantum counterparts. In other words, we consider the inverse Radon transform of the exact quantum probability distributions of all linear combinations. We call it the Wigner distribution because for position and momentum, this property defines the standard Wigner function. We discuss the application to finite dimensional systems, establish many basic properties, and illustrate these by examples. The properties include the support, the location of singularities, positivity, the behavior under symmetry groups, and informational completeness.
In this note, we analyze joint probability distributions that arise from outcomes of sequences of quantum measurements performed on sets of quantum states. First, we identify some properties of these distributions that need to be fulfilled to get a classical behavior. Secondly, we prove that a joint distribution exists iff measurement operators "on-state" permute (permutability is the commutativity of more than two operators). By "on-state" we mean properties of operators that hold only on a subset of states in the Hilbert space. Then, we disprove a conjecture proposed by Carstens, Ebrahimi, Tabia, and Unruh (eprint 2018), which states that the property of partial on-state permutation implies full on-state permutation. We disprove this conjecture with a counterexample where pairwise "on-state" commutativity does not imply on-state permutability, unlike in the case of commutativity for all states in the Hilbert space. Finally, we explore the new concept of on-state commutativity by showing a simple proof that if two projections almost on-state commute, then there is a commuting pair of operators that are on-state close to the originals. This result was originally proven by Hastings (Communications in Mathematical Physics, 2019) for general operators.