scholarly journals Compatibility for probabilistic theories

2016 ◽  
Vol 66 (2) ◽  
Author(s):  
Stan Gudder
Keyword(s):  
Probability Theory ◽  
Quantum Mechanics ◽  
Quantum Logic ◽  
Probabilistic Theory ◽  
Classical Probability ◽  
Classical Probability Theory ◽  
Positive Vector ◽  
Vector Valued ◽  
Pt Quantum Mechanics ◽  
Probabilistic Theories

AbstractWe define an index of compatibility for a probabilistic theory (PT). Quantum mechanics with index 0 and classical probability theory with index 1 are at the two extremes. In this way, quantum mechanics is at least as incompatible as any PT. We consider a PT called a concrete quantum logic that may have compatibility index strictly between 0 and 1, but we have not been able to show this yet. Finally, we show that observables in a PT can be represented by positive, vector-valued measures.

scholarly journals Probabilistic frames for non-Boolean phenomena

2016 ◽  
Vol 374 (2058) ◽  
pp. 20150102 ◽  
Author(s):  
Louis Narens
Keyword(s):  
Probability Theory ◽  
Quantum Mechanics ◽  
Quantum Probability ◽  
Classical Probability ◽  
Classical Probability Theory ◽  
Foundations Of Probability ◽  
Quantifying Uncertainty ◽  
Behavioural Sciences ◽  
Generalized Probability ◽  
Provided Examples

Classical probability theory, as axiomatized in 1933 by Andrey Kolmogorov, has provided a useful and almost universally accepted theory for describing and quantifying uncertainty in scientific applications outside quantum mechanics. Recently, cognitive psychologists and mathematical economists have provided examples where classical probability theory appears inadequate but the probability theory underlying quantum mechanics appears effective. Formally, quantum probability theory is a generalization of classical probability. This article explores relationships between generalized probability theories, in particular quantum-like probability theories and those that do not have full complementation operators (e.g. event spaces based on intuitionistic logic), and discusses how these generalizations bear on important issues in the foundations of probability and the development of non-classical probability theories for the behavioural sciences.

scholarly journals Quantum Metrics for Continuous Shape Measures of Molecules

2020 ◽  
Author(s):  
Andrei Tchougreeff
Keyword(s):  
Probability Theory ◽  
Quantum Mechanics ◽  
Numerical Algorithms ◽  
Computer Code ◽  
Molecular Shape ◽  
Continuous Symmetry ◽  
Classical Probability ◽  
Classical Probability Theory ◽  
Vertex Set

The field of continuous molecular shape and symmetry (dis)similarity quantifiers habitually called measures (specifically continuous shape measures - CShM or continuous symmetry measures - CSM) is obfuscated by the combinatorial numerical algorithms used in the field which restricts the applicability to the molecules containing up to twenty equivalent atoms. In the present paper we analyze this problem using various tools of classical probability theory as well as of one-particle and many-particle quantum mechanics. Applying these allows us to lift the combinatorial restriction and to identify the adequate renumbering of atoms (vertices) without considering all N! permutations of an N-vertex set so that in the end purely geometric molecular shape (dis)similarity quantifier can be defined. Developed methods can be easily implemented in the relevant computer code.<br>

scholarly journals Probabilistic theories and reconstructions of quantum theory

2021 ◽  
Author(s):  
Markus Müller
Keyword(s):  
Probability Theory ◽  
Quantum Theory ◽  
Three Dimensional ◽  
Complex Numbers ◽  
Classical Probability ◽  
Quantum Bit ◽  
Classical Probability Theory ◽  
Generalized Probabilistic Theories ◽  
Usual Representation ◽  
Probabilistic Theories

These lecture notes provide a basic introduction to the framework of generalized probabilistic theories (GPTs) and a sketch of a reconstruction of quantum theory (QT) from simple operational principles. To build some intuition for how physics could be even more general than quantum, I present two conceivable phenomena beyond QT: superstrong nonlocality and higher-order interference. Then I introduce the framework of GPTs, generalizing both quantum and classical probability theory. Finally, I summarize a reconstruction of QT from the principles of Tomographic Locality, Continuous Reversibility, and the Subspace Axiom. In particular, I show why a quantum bit is described by a Bloch ball, why it is three-dimensional, and how one obtains the complex numbers and operators of the usual representation of QT.

scholarly journals Quantum Metrics for Continuous Shape Measures of Molecules

2020 ◽  
Author(s):  
Andrei Tchougreeff
Keyword(s):  
Probability Theory ◽  
Quantum Mechanics ◽  
Numerical Algorithms ◽  
Computer Code ◽  
Molecular Shape ◽  
Continuous Symmetry ◽  
Classical Probability ◽  
Classical Probability Theory ◽  
Vertex Set

The field of continuous molecular shape and symmetry (dis)similarity quantifiers habitually called measures (specifically continuous shape measures - CShM or continuous symmetry measures - CSM) is obfuscated by the combinatorial numerical algorithms used in the field which restricts the applicability to the molecules containing up to twenty equivalent atoms. In the present paper we analyze this problem using various tools of classical probability theory as well as of one-particle and many-particle quantum mechanics. Applying these allows us to lift the combinatorial restriction and to identify the adequate renumbering of atoms (vertices) without considering all N! permutations of an N-vertex set so that in the end purely geometric molecular shape (dis)similarity quantifier can be defined. Developed methods can be easily implemented in the relevant computer code.<br>

scholarly journals Quantum mechanics as a generalized probability theory

2020 ◽  
Author(s):  
William Icefield
Keyword(s):  
Probability Theory ◽  
Quantum Mechanics ◽  
Quantum Probability ◽  
Classical Probability ◽  
Classical Probability Theory ◽  
Principle Of Maximum Entropy ◽  
Principle Of Indifference ◽  
Reconstruction Project ◽  
Principle Of Maximum ◽  
Generalized Probability

When quantum mechanics is understood as a new generalized theory of probability - to be called the quantum probability theory - mysteries and controversies regarding quantum mechanics are dissolved. In the classical probability theory, that a measurement of some system requires an additional measurement apparatus is of insignificant importance - in the quantum probability theory, this comes to change. For one central single reason around a particular classical probability equation, the generalized probability view has not gained much traction, despite the fact that this essentially echoes (and provides logical underpinnings of) the conventional wisdom that `quantum mechanics just works as it is.' A classical probability axiom is just an initial intuition - there is no reason why we have to dogmatically cling onto axioms that can clearly be generalized. Issues with the principle of indifference in the classical probability theory are emphasized, along with the quantum reconstruction project of deriving quantum mechanics from epistemic requirements and potential quantum gravity consequences from the principle of maximum entropy.

scholarly journals SOME IDENTITIES FOR THE QUANTUM MEASURE AND ITS GENERALIZATIONS

2002 ◽  
Vol 17 (12) ◽  
pp. 711-728 ◽  
Author(s):  
ROBERTO B. SALGADO
Keyword(s):  
Probability Theory ◽  
Quantum Mechanics ◽  
Sum Rules ◽  
Measure Theory ◽  
Classical Probability ◽  
Sum Rule ◽  
Classical Probability Theory ◽  
Quantum Measure ◽  
Open Questions ◽  
Special Case

After a brief review of classical probability theory (measure theory), we present an observation (due to Sorkin) concerning an aspect of probability in quantum mechanics. Following Sorkin, we introduce a generalized measure theory based on a hierarchy of "sum-rules". The first sum-rule yields classical probability theory, and the second yields a generalized probability theory that includes quantum mechanics as a special case. We present some algebraic relations involving these sum-rules. This may be useful for the study of the higher-order sum-rules and possible generalizations of quantum mechanics. We conclude with some open questions and suggestions for further work.

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