Joint Distributions and Quantum Nonlocal Models

Malley, James D.; Fletcher, Anthony R.

doi:10.3390/axioms3020166

Cited by 4 publications

(5 citation statements)

References 8 publications

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“…The agreement of these four marginals with observation can imply, or disallow, a classical probability distribution for the pair of outcomes. This has been examined in [20]. As noted following Lemma 1, no versions, conditions, or properties of quantum conditional probability and any links to classical conditional property are invoked or required for the results obtained here.…”

Section: Discussionmentioning

confidence: 99%

Classical Probability and Quantum Outcomes

Malley

2014

Axioms

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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.

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Section: Discussionmentioning

confidence: 99%

Classical Probability and Quantum Outcomes

Malley

2014

Axioms

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“…Let B = {B j } be another POVM for S. Given post measurement state ρ i , the probability of measurement with element B j resulting in value b j is, by recursively applying (5), p(b j |ρ i ) = Tr(ρ i B j ). Substituting the expression for ρ i in (6), I obtain the conditional probability…”

Section: The Law Of Total Probability In Quantum Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Law of Total Probability in Quantum Theory and Its Application in Wigner’s Friend Scenario

Yang

2022

Entropy

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It is well-known that the law of total probability does not generally hold in quantum theory. However, recent arguments on some of the fundamental assumptions in quantum theory based on the extended Wigner’s friend scenario show a need to clarify how the law of total probability should be formulated in quantum theory and under what conditions it still holds. In this work, the definition of conditional probability in quantum theory is extended to POVM measurements. A rule to assign two-time conditional probability is proposed for incompatible POVM operators, which leads to a more general and precise formulation of the law of total probability. Sufficient conditions under which the law of total probability holds are identified. Applying the theory developed here to analyze several quantum no-go theorems related to the extended Wigner’s friend scenario reveals logical loopholes in these no-go theorems. The loopholes exist as a consequence of taking for granted the validity of the law of total probability without verifying the sufficient conditions. Consequently, the contradictions in these no-go theorems only reconfirm the invalidity of the law of total probability in quantum theory rather than invalidating the physical statements that the no-go theorems attempt to refute.

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“…Let B = {B j } is another POVM for S. Given the post measurement state ρ i , the probability of measurement with element B j resulting in value b j is, by applying recursively (5), p(b j |ρ i ) = T r(ρ i B j ). Substituting the expression for ρ i in (6), we obtain the conditional probability…”

Section: The Law Of Total Probability In Quantum Theorymentioning

confidence: 99%

“…Indeed, many other classical probability rules are only upheld in specific conditions. For instance, a joint probability can be definitely assigned only when the two measurement operators are commutative [3][4][5][6]; There are many variants of definition of the conditional probability in quantum theory (see a review in [7]). However, a family of no-go theorems recently * jianhao.yang@alumni.utoronto.ca published [12,15,18,19] appear to rely on the total law of probability one way or another without considering the sufficient conditions.…”

Section: Introductionmentioning

confidence: 99%

Law of Total Probability in Quantum Theory and Its Application in Wigner's Friend Scenario

Yang

2022

Preprint

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It is well-known that the law of total probability does not hold in general in quantum theory. However, the recent arguments on some of the fundamental assumptions in quantum theory based on the extended Wigner's Friend scenario show a need to clarify how the law of total probability should be formulated in quantum theory and under what conditions it still holds. In this work, the definition of conditional probability in quantum theory is extended to POVM measurements. Rule to assign two-time conditional probability is proposed for incompatible POVM operators, which leads to a more general and precise formulation of the law of total probability. Sufficient conditions under which the law of total probability holds are identified. Applying the theory developed here to analyze several quantum no-go theorems related to the extended Wigner's friend scenario reveals logical loopholes in these no-go theorems. The loopholes exist as a consequence of taking for granted the validity of the law of total probability without verifying the sufficient conditions. Consequently, the contradictions in these no-go theorems just reconfirm the invalidity of the law of total probability in quantum theorem, instead of due to the desired statements.

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Joint Distributions and Quantum Nonlocal Models

Cited by 4 publications

References 8 publications

Classical Probability and Quantum Outcomes

Classical Probability and Quantum Outcomes

Law of Total Probability in Quantum Theory and Its Application in Wigner’s Friend Scenario

Law of Total Probability in Quantum Theory and Its Application in Wigner's Friend Scenario

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