The quantum probability calculus

Synthese ◽  
1974 ◽  
Vol 29 (1-4) ◽  
pp. 131-154 ◽  
Author(s):  
J. M. Jauch
Keyword(s):  
Quantum Probability ◽  
Probability Calculus

Quantum Probability Calculus as Fuzzy-Kolmogorovian Probability Calculus

2009 ◽  
Author(s):  
Jarosław Pykacz ◽  
Luigi Accardi ◽  
Guillaume Adenier ◽  
Christopher Fuchs ◽  
Gregg Jaeger ◽  
...  
Keyword(s):  
Quantum Probability ◽  
Probability Calculus

HILBERT SPACE OVER COMPLEX HYPERBOLIC NUMBERS AND HYPER-TRIGONOMETRIC INTERFERENCE

2009 ◽  
Vol 12 (03) ◽  
pp. 469-478
Author(s):  
A. KHRENNIKOV
Keyword(s):  
Hilbert Space ◽  
Quantum Probability ◽  
Complex Hilbert Space ◽  
Hyperbolic Numbers ◽  
Probability Calculus ◽  
Interference Of Probabilities

This note is devoted to extension of quantum probability calculus to generalizations of complex Hilbert space. Starting with Hilbert space over complex hyperbolic numbers, we derive general hyper-trigonometric interference of probabilities.

BELL-TYPE INEQUALITIES IN FUZZY PROBABILITY CALCULUS

2001 ◽  
Vol 09 (02) ◽  
pp. 263-275 ◽  
Author(s):  
JAROSŁAW PYKACZ ◽  
BART D'HOOGHE
Keyword(s):  
Fuzzy Set ◽  
Mathematical Physics ◽  
Quantum Probability ◽  
Physical Situation ◽  
Fuzzy Probability ◽  
Probability Models ◽  
Triangular Norms ◽  
Probability Calculus ◽  
Standard Set ◽  
Fuzzy Probabilities

Bell-type inequalities, used in mathematical physics as a criterion to check whether a physical situation allows description in terms of classical (Kolmogorovian) or quantum probability calculus are applied to various fuzzy probability models. It occurs that the standard set of Bell-type inequalities does not allow to distinguish Kolmogorovian probabilities from fuzzy probabilities based on the most frequently used Zadeh intersection or probabilistic intersection, but it allows to distinguish all these models from fuzzy probability models based on Giles (Łukasiewicz) intersection. It is proved that if we use fuzzy set intersections pointwisely generated by Frank's fundamental triangular norms Ts(x,y), then the borderline between fuzzy probability models that can be distinguished from Kolmogorovian ones and these fuzzy probability models that cannot be distinguished is for [Formula: see text].

An Invitation to Quantum Probability Calculus

2020 ◽  
pp. 121-140
Author(s):  
Hung T. Nguyen ◽  
Nguyen Duc Trung ◽  
Nguyen Ngoc Thach
Keyword(s):  
Quantum Probability ◽  
Probability Calculus

SPIN-PARITY EFFECT IN VIOLATION OF BELL'S INEQUALITIES

2013 ◽  
Vol 28 (01) ◽  
pp. 1450004 ◽  
Author(s):  
ZHIGANG SONG ◽  
J.-Q. LIANG ◽  
L.-F. WEI
Keyword(s):  
Berry Phase ◽  
Quantum Probability ◽  
Quantum Nonlocality ◽  
Entangled State ◽  
Bell’S Inequalities ◽  
Bipartite Entanglement ◽  
Spin Parity ◽  
Parity Effect ◽  
Bell's Inequalities ◽  
Spin Parity Effect

Analytic formulas of Bell correlations are derived in terms of quantum probability statistics under the assumption of measuring outcome-independence and the Bell's inequalities (BIs) are extended to general bipartite-entanglement macroscopic quantum-states (MQS) of arbitrary spins. For a spin-½ entangled state we find analytically that the violations of BIs really resulted from the quantum nonlocal correlations. However, the BIs are always satisfied for the spin-1 entangled MQS. More generally the quantum nonlocality does not lead to the violation for the integer spins since the nonlocal interference effects cancel each other by the quantum statistical-average. Such a cancellation no longer exists for the half-integer spins due to the nontrivial Berry phase, and thus the violation of BIs is understood remarkably as an effect of geometric phase. Specifically, our generic observation of the spin-parity effect can be experimentally tested with the entangled photon-pairs.

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