Is complex probability theory consistent with Bell's theorem?

Youssef, S.

doi:10.1016/0375-9601(95)00467-h

Cited by 12 publications

(7 citation statements)

References 33 publications

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“…A complex probability space is a triple (X, Σ, µ), where Σ is an algebra for the nonempty set X and µ : Σ → C is such that Just as with negative probability spaces, complex probability spaces differ from classical probability spaces only in the range of the measure. Youssef shows that complex probability spaces arise from a generalization of the Cox (1946) axioms for probability that, among other virtues, explicitly allow one to avoid the standard conclusions of Bell's theorem against local, realistic theories (Youssef, 1994(Youssef, , 1995. Miller also arrives at complex probability spaces through a consideration foundational issues, in particular of time-symmetric formulations of quantum mechanics involving weak measurement.…”

Section: Extended Probability Spacesmentioning

confidence: 99%

On Noncontextual, Non-Kolmogorovian Hidden Variable Theories

Feintzeig

Fletcher

2017

Found Phys

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One implication of Bell's theorem is that there cannot in general be hidden variable models for quantum mechanics that both are noncontextual and retain the structure of a classical probability space. Thus, some hidden variable programs aim to retain noncontextuality at the cost of using a generalization of the Kolmogorov probability axioms. We generalize a theorem of Feintzeig (2015) to show that such programs are committed to the existence of a finite null cover for some quantum mechanical experiments, i.e., a finite collection of probability zero events whose disjunction exhausts the space of experimental possibilities.

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Section: Extended Probability Spacesmentioning

confidence: 99%

On Noncontextual, Non-Kolmogorovian Hidden Variable Theories

Feintzeig

Fletcher

2017

Found Phys

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“…Thus one possible way to avoid these no-go results is to relax one or more of the classical probability axioms, leading to a non-classical probability space in which it is possible to come up with a non-contextual model. A number of such proposals have been made, including extended probability spaces which allow negative probabilities [2] or complex probabilities [53][54][55], generalized probability spaces [56] which relax the requirement that we should be able to assign probabilities to all conjunctions of events, upper probability spaces which are subadditive rather than additive on disjoint measurable sets [57], and quantum measures which are not additive on pairs of events but which are additive on triples of events [58].…”

Section: Generalisations Of Probabilitymentioning

confidence: 99%

Contextuality, Fine-Tuning and Teleological Explanation

Adlam

2021

Found Phys

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I assess various proposals for the source of the intuition that there is something problematic about contextuality, ultimately concluding that contextuality is best thought of in terms of fine-tuning. I then argue that as with other fine-tuning problems in quantum mechanics, this behaviour can be understood as a manifestation of teleological features of physics. Finally I discuss several formal mathematical frameworks that have been used to analyse contextuality and consider how their results should be interpreted by scientific realists. In the course of this discussion I obtain several new mathematical results-I demonstrate that preparation contextuality is a form of fine-tuning, I show that measurement contextuality can be explained by appeal to a global constraint forbidding closed causal loops, and I demonstrate how negative probabilities can arise from a classical ontological model together with an epistemic restriction.

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“…. This is an interesting point because it is exactly this failure that allows complex probability theory to escape Bell's theorem [3]. Also, although I don't have a sharp result, it is seems likely that this effect disappears in a classical limit, thus explaining why standard probability theory works in the classical domain.…”

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confidence: 95%

“…This suggests that perhaps there is something wrong with probability theory after all, and that this may be where quantum mechanical effects come from. Let's adopt this point of view and see where it leads [1,2,3].…”

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confidence: 99%

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Is Quantum Mechanics an Exotic Probability Theory?

Youssef¹

1995

Annals of the New York Academy of Sciences

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Quantum mechanics is often introduced with a discussion of the two-slit experiment where the observed interference pattern forces us to conclude that the particle goes through both slits at once and is "both a particle and a wave". This most-basic argument has a loophole, however. The conclusion rests on probability theory and, in particular, on the fact that probabilities are nonnegative so that, when the second slit is opened, P(x) = P(x via slit 1) + P(x via slit 2) 2 P(x via slit l),where P(x) is the probability for the particle to arrive at position x on the screen. Perhaps, then, it is possible that the particle does go through either one slit or the other after all and the interference effects can be explained by modifying probability theory itself. Here, we give a brief overview of this program, which is put forward in references 1-3. For other approaches to exotic probability theory, see references 4-7.Probability theory is most often presented with the "frequentist" approach where probabilities are defined as limits of experimental frequencies. For example, if n successes occur in N trials of an experiment, the large N limit of n/N is called the "probability of success". Such probabilities are then assumed to follow Kolmogorov's axioms8 From this point of view, probability theory is a theory of "random phenomena" that are well described by these axioms and, of course, probabilities are necessarily nonnegative. For our purposes, then, it is necessary to adopt the more general "Bayesian" view where probabilities are not a priori defined as frequencies, but where, instead, a frequency interpretation is derived as a consequence of the fundamental axiom^.^ With the Bayesian view, probability is introduced as a measure of "likelihood" where, if proposition a is known, the likelihood that proposition b is true is denoted1° "(a + b)". For "+" to be a useful likelihood measure, one expects it to have a few properties such as (i) if (a + b ) is known, this should determine (a + 7 b ) and (ii) the procedure to get from (a + b) to (a + T b ) should be independent of a and b. As shown by COX,^^ such considerations are enough to entirely fix probability theory, which then takes the form (a + b A c ) = (a + b)(a A b + c ) (a + b ) + (a + T b ) = 1 (a + -a ) = 0 (2a) (2b) (2c)for all propositions a, b, and c. 904

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Is complex probability theory consistent with Bell's theorem?

Abstract: Bayesian complex probability theory is shown to be consistent with Bell's theorem and with other recent limitations on local realistic theories which agree with the predictions of quantum mechanics.

Cited by 12 publications

References 33 publications

On Noncontextual, Non-Kolmogorovian Hidden Variable Theories

On Noncontextual, Non-Kolmogorovian Hidden Variable Theories

Contextuality, Fine-Tuning and Teleological Explanation

Is Quantum Mechanics an Exotic Probability Theory?

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