Indistinguishability and Negative Probabilities

Barros, J. Acácio de; Holik, Federico

doi:10.3390/e22080829

Cited by 10 publications

(14 citation statements)

References 72 publications

(103 reference statements)

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“…More recently, negative probabilities have been explored in various quantum mechanical contexts, such as indistiguishability [24], quantum computation [25], and contextuality [26]. Besides, an operational interpretation of negative probabilities has been proposed by Abramsky and Brandenburger [27,28].…”

Section: Quantum Mechanicsmentioning

confidence: 99%

Logical entropy and negative probabilities in quantum mechanics

Manfredi

2022

4open

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The concept of Logical Entropy, SL = 1-Σni=1 pi2, where the pi are normalized probabilities, was introduced by David Ellerman in a series of recent papers. Although the mathematical formula itself is not new, Ellerman provided a sound probabilistic interpretation of SL as a measure of the distinctions of a partition on a given set. The same formula comes across as a useful definition of entropy in quantum mechanics, where it is linked to the notion of purity of a quantum state. The quadratic form of the logical entropy lends itself to a generalization of the probabilities that include negative values, an idea that goes back to Feynman and Wigner. Here, we analyze and reinterpret negative probabilities in the light of the concept of logical entropy. Several intriguing quantum-like properties of the logical entropy are derived and discussed in finite dimensional spaces. For infinite-dimensional spaces (continuum), we show that, under the sole hypothesis that the logical entropy and the total probability are preserved in time, one obtains an evolution equation for the probability density that is basically identical to the quantum evolution of the Wigner function in phase space, at least when one considers only the momentum variable. This result suggest that the logical entropy plays a profound role in establishing the peculiar rules of quantum physics.

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Section: Quantum Mechanicsmentioning

confidence: 99%

Logical entropy and negative probabilities in quantum mechanics

Manfredi

2022

4open

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“…Not surprisingly, extended probability is related with contextuality. The difference between the total sum of the probability's modulus and 1 (which is, of course, the total sum in classical probability) defines a coefficient of contextuality [24,25]. Other alternative "quasi-Boolean" descriptions of Bell's experiment involve probabilities defined over singular measures [26] or non-measurable sets [27], or p-adic ultrametric distances [28][29][30].…”

Section: Is It Always Impossible Applying Probabilities To Non-boolea...mentioning

confidence: 99%

Non-Boolean hidden variable model reproduces Quantum Mechanics’ predictions for Bell’s experiment.

Hnilo¹

2022

Preprint

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The experimentally verified violation of Bell’s inequalities apparently implies that at least one of two beliefs, which are accepted as true in almost all scientific practice, must be false. These beliefs are: that effects propagating at infinite velocity do not exist, and that natural phenomena occur independently of being observed. Giving up any one of these two beliefs (usually known together as Local Realism) is controversial. For many years, many theories have been proposed to reconcile the violation of Bell’s inequalities with Local Realism, but none has been fully successful. In this paper, it is recalled that Bell’s inequalities are equivalent to the conditions to decide the completeness of a theory according to Boolean logic. Therefore, any theory aimed to violate Bell’s inequalities must start by giving up Boolean logic. Two problems are therefore defined: the “soft” one is to explain the violation of Bell’s inequalities without violating (non-Boolean) Local Realism; the “hard” one is to predict the time values when single particles are detected, in such a way that the resulting number of coincidences violates Bell’s inequalities. A simple hidden variables model is introduced, which solves the “soft” problem even in an ideal setup. This is possible thanks to the use of vectors in real space as the hidden variables and the corresponding operation (projection), which do not hold to Boolean logic. This model reconciles the violation of Bell’s inequalities with Local Realism and should end a controversy that has lasted for decades. Regarding the “hard” problem, the introduced model is as incomplete as Quantum Mechanics is. It is claimed that solving the “hard” problem requires devising a new kind of quantum computer, which should be able to accept (non-Boolean) hidden variable values as input data and to replace the statistical Born’s rule of Quantum Mechanics with a deterministic threshold condition.

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“…Random variables with the same content (but belonging to different contexts) are made equal without further discussion. In a recent series of works [ 58 , 61 , 62 ], it was proposed that repeated experiments are neither equal nor different. They are considered indistinguishable , the latter being a notion taken from non-reflexive logic.…”

Section: The Main Features Of Quantum Probabilitymentioning

confidence: 99%

Non-Kolmogorovian Probabilities and Quantum Technologies

Holik

2022

Entropy

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In this work, we focus on the philosophical aspects and technical challenges that underlie the axiomatization of the non-Kolmogorovian probability framework, in connection with the problem of quantum contextuality. This fundamental feature of quantum theory has received a lot of attention recently, given that it might be connected to the speed-up of quantum computers—a phenomenon that is not fully understood. Although this problem has been extensively studied in the physics community, there are still many philosophical questions that should be properly formulated. We analyzed different problems from a conceptual standpoint using the non-Kolmogorovian probability approach as a technical tool.

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Indistinguishability and Negative Probabilities

Cited by 10 publications

References 72 publications

Logical entropy and negative probabilities in quantum mechanics

Logical entropy and negative probabilities in quantum mechanics

Non-Boolean hidden variable model reproduces Quantum Mechanics’ predictions for Bell’s experiment.

Non-Kolmogorovian Probabilities and Quantum Technologies

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