Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Axioms

Joint Distributions and Quantum Nonlocal Models

James D. Malley, Anthony R. Fletcher

Published Apr 15, 2014• 5 citations
Publisher

Abstract

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.

DOI
10.3390/axioms3020166

Related papers

Classical Probability and Quantum Outcomes

James D. Malley

There is a contact problem between classical probability and quantum outcomes. Thus, a standard result from classical probability on the existence of joint distributions ultimately implies that all quantum observables must commute. An essential task here is a closer identification of this conflict based on deriving commutativity from the weakest possible assumptions, and showing that stronger assumptions in some of the existing no-go proofs are unnecessary. An example of an unnecessary assumption in such proofs is an entangled system involving nonlocal observables. Another example involves the Kochen-Specker hidden variable model, features of which are also not needed to derive commutativity. A diagram is provided by which user-selected projectors can be easily assembled into many new, graphical no-go proofs.

Axioms•May 26, 2014•1 citation

Quantum Joint Distributions

Todd A. Oliynyk

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.

arXiv:quant-ph/0304140•Apr 22, 2003•0 citations

Joint distributions, the uncertainty principle and positive distributions

Leon W. Cohen

We examine the construction of joint probabilities for non-commuting observables. We show that there are indications in standard quantum mechanics that imply the existence of conditional expectation values, which in turn implies the existence of a joint distribution. We also argue that the uncertainty principle has no bearing on the existence of joint distributions but only constrains the marginal distributions. In addition, we show that within classical probability theory there are mathematical quantities that are similar to quantum mechanical wave functions. This is shown by generalising a theorem of Khinchin on the necessary and sufficient conditions for a function to be a characteristic function.

Mathematical Structures in Computer Science•Mar 28, 2014•0 citations

Is Bell’s Theorem Relevant to Quantum Mechanics? On Locality and Non-Commuting Observables

A. Matzkin, Luigi Accardi, Guillaume Adenier +5 more

Bell's theorem is a statement by which averages obtained from specific types of statistical distributions must conform to a family of inequalities. These models, in accordance with the EPR argument, provide for the simultaneous existence of quantum mechanically incompatible quantities. We first recall several contradictions arising between the assumption of a joint distribution for incompatible observables and the probability structure of quantum-mechanics, and conclude that Bell's theorem is not expected to be relevant to quantum phenomena described by non-commuting observables, irrespective of the issue of locality. Then, we try to disentangle the locality issue from the existence of joint distributions by introducing two models accounting for the EPR correlations but denying the existence of joint distributions. We will see that these models do not need to resort explicitly to non-locality: the first model relies on conservation laws for ensembles, and the second model on an equivalence class by which different configurations lead to the same physical predictions.

arXiv:0802.0613•AIP conference proceedings•Jan 1, 2009•8 citations

Quantum non-locality co-exists with locality

ARMEN E. ALLAHVERDYAN, Arshag Danageozian

Quantum non-locality is normally defined via violations of Bell's inequalities that exclude certain classical hidden variable theories from explaining quantum correlations. Another definition of non-locality refers to the wave function collapse thereby one can prepare a quantum state from arbitrary far away. In both cases one can debate on whether non-locality is a real physical phenomenon, e.g. , one can employ formulations of quantum mechanics that do not use collapse, or one can simply refrain from explaining quantum correlations via classical hidden variables. Here we point out that there is a non-local effect within quantum mechanics, i.e. , without involving hidden variables or collapse. This effect is seen via imprecise ( i.e. , interval-valued) joint probability of two observables, which replaces the ill-defined notion of the precise joint probability for non-commuting observables. It is consistent with all requirements for the joint probability, e.g. , those for commuting observales. The non-locality amounts to a fact that (in a two-particle system) the joint imprecise probability of non-commuting two-particle observables ( i.e. , tensor product of single-particle observables) does not factorize into single-particle contributions, even for uncorrelated states of the two-particle system. The factorization is recovered for a less precise ( i.e. , the one involving a wider interval) joint probability. This approach to non-locality reconciles it with locality, since the latter emerges as a less precise description.

arXiv:1807.02852•Europhysics Letters (EPL)•Jul 2, 2018•2 citations

On-State Commutativity of Measurements and Joint Distributions of Their Outcomes

Jan Czajkowski, Alex B. Grilo

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.

arXiv:2101.08313•Jan 20, 2021•0 citations
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.
There is a contact problem between classical probability and quantum outcomes. Thus, a standard result from classical probability on the existence of joint distributions ultimately implies that all quantum observables must commute. An essential task here is a closer identification of this conflict based on deriving commutativity from the weakest possible assumptions, and showing that stronger assumptions in some of the existing no-go proofs are unnecessary. An example of an unnecessary assumption in such proofs is an entangled system involving nonlocal observables. Another example involves the Kochen-Specker hidden variable model, features of which are also not needed to derive commutativity. A diagram is provided by which user-selected projectors can be easily assembled into many new, graphical no-go proofs.
We examine the construction of joint probabilities for non-commuting observables. We show that there are indications in standard quantum mechanics that imply the existence of conditional expectation values, which in turn implies the existence of a joint distribution. We also argue that the uncertainty principle has no bearing on the existence of joint distributions but only constrains the marginal distributions. In addition, we show that within classical probability theory there are mathematical quantities that are similar to quantum mechanical wave functions. This is shown by generalising a theorem of Khinchin on the necessary and sufficient conditions for a function to be a characteristic function.
Bell's theorem is a statement by which averages obtained from specific types of statistical distributions must conform to a family of inequalities. These models, in accordance with the EPR argument, provide for the simultaneous existence of quantum mechanically incompatible quantities. We first recall several contradictions arising between the assumption of a joint distribution for incompatible observables and the probability structure of quantum-mechanics, and conclude that Bell's theorem is not expected to be relevant to quantum phenomena described by non-commuting observables, irrespective of the issue of locality. Then, we try to disentangle the locality issue from the existence of joint distributions by introducing two models accounting for the EPR correlations but denying the existence of joint distributions. We will see that these models do not need to resort explicitly to non-locality: the first model relies on conservation laws for ensembles, and the second model on an equivalence class by which different configurations lead to the same physical predictions.
Quantum non-locality is normally defined via violations of Bell's inequalities that exclude certain classical hidden variable theories from explaining quantum correlations. Another definition of non-locality refers to the wave function collapse thereby one can prepare a quantum state from arbitrary far away. In both cases one can debate on whether non-locality is a real physical phenomenon, e.g. , one can employ formulations of quantum mechanics that do not use collapse, or one can simply refrain from explaining quantum correlations via classical hidden variables. Here we point out that there is a non-local effect within quantum mechanics, i.e. , without involving hidden variables or collapse. This effect is seen via imprecise ( i.e. , interval-valued) joint probability of two observables, which replaces the ill-defined notion of the precise joint probability for non-commuting observables. It is consistent with all requirements for the joint probability, e.g. , those for commuting observales. The non-locality amounts to a fact that (in a two-particle system) the joint imprecise probability of non-commuting two-particle observables ( i.e. , tensor product of single-particle observables) does not factorize into single-particle contributions, even for uncorrelated states of the two-particle system. The factorization is recovered for a less precise ( i.e. , the one involving a wider interval) joint probability. This approach to non-locality reconciles it with locality, since the latter emerges as a less precise description.
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.