On the characterization of probabilities: A generalization of Bell’s inequalities

Beltrametti, Enrico G.; Mączyński, Maciej

doi:10.1063/1.530333

Cited by 36 publications

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“…Let S be a set of states of a physical system and p(s) the probability of an occurrence of an event when the system is in state s ∈ S. The function p from S to [0, 1] is called a numerical event, multidimensional probability or, more precisely, an S-probability (cf. [3], [4]). We note that p(s) can also be considered as a special case of Mackey's probability function p(A, s, E), with A a fixed observable, s a variable state, and E a fixed Borel set (cf.…”

Section: Introductionmentioning

confidence: 99%

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Boolean Properties and Bell-Like Inequalities of Numerical Events

Dorninger

Länger

Mączyński

2020

Reports on Mathematical Physics

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Let S be a set of states of a physical system and p(s) be the probability of the occurrence of an event when the system is in state s ∈ S. A function p : S → [0, 1] is called a numerical event or alternatively an S-probability. If a set P := {p(s) | s ∈ S} is ordered by the order of real functions such that certain plausible requirements are fulfilled, P becomes an orthomodular poset in which properties can be described by the addition and comparison of functions. P is then called an algebra of S-probabilities or algebra of numerical events. We first answer the question under which circumstances it is possible to consider sets of empirically found numerical events as members of an algebra of S-probabilities. Then we discuss the problem to decide whether a given small set P n of S-probabilities can be embedded into a Boolean subalgebra of an algebra P of S-probabilities, in which case we will call P n Boolean embeddable. If P n is not Boolean embeddable, then the physical system at hand will most likely be non-classical. In the case of a concrete logic P , that is a quantum logic which can be represented by sets, we derive criteria for P n ⊆ P to be Boolean embeddable which can be checked by very simple procedures, for arbitrary S-probabilities we provide sets of Bell-like inequalities which characterize the Boolean embeddability of P n . Finally we will show how these Bell-like inequalities fit into a general framework of Bell inequalities by providing a method for generating Bell inequalities for S-probabilities from elementary Bell valuations.

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Section: Introductionmentioning

confidence: 99%

“…We further agree that the symbols + and − indicate the addition and subtraction in R. Then a set P of S-probabilities is called a space of numerical events (cf. [4]) or an algebra of S-probabilities (or an algebra of numerical events) (first mentioned in [2]), if it satisfies the following axioms:…”

Section: Introductionmentioning

confidence: 99%

Boolean Properties and Bell-Like Inequalities of Numerical Events

Dorninger

Länger

Mączyński

2020

Reports on Mathematical Physics

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“…1 This is particularly evident in the quantum logic approach where several theorems were proved showing that various versions of Bell-type inequalities are satisfied by probability measures defined on a quantum logic if this logic is a Boolean algebra (see, e.g. papers by Santos [2], Pulmannová and Majernik [3], or Beltrametti and Maczyński [4,5]). …”

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confidence: 95%

Perspectives

Pykacz¹

2015

SpringerBriefs in Physics

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PerspectivesIsomorphic representation of Birkhoff-von Neumann quantum logics, and therefore also of orthomodular lattices L(H) of (orthogonal projections onto) closed subspaces of Hilbert spaces by families of fuzzy sets endowed with Łukasiewicz operations opens new opportunities for solving at least two long-standing problems, namely the development of quantum probability calculus in a way completely analogous to the orthodox Kolmogorovian probability theory, and the construction of a phase space representation of quantum mechanics not plagued by the appearance of negative probabilities. However, it should be stressed that what we present here is only a brief prospect for future studies which will certainly require much further investigation. Fuzzy Set Models of Quantum ProbabilityIn some experiments on quantum systems the relative frequencies of obtaining various results, interpreted as probabilities, do not fulfil the numerical constraints imposed by classical (Kolmogorovian) probability theory. Such instances, usually connected with the violation of Bell's inequalities, strongly indicate the necessity of modification of the probability calculus used in quantum mechanics.There are several approaches to the subject that can generally be termed "quantum probability" and even the brief review of all of these would lead us far beyond the scope of this section. Therefore, we shall concentrate on the quantum-logical treatment of this subject.In the quantum logic approach to the foundations of quantum mechanics the Kolmogorovian triple ( , F, P) consisting of a space of elementary events , a Boolean σ-algebra F of selected subsets of (random events), and a probability measure P, is replaced by a couple (L , p) consisting of a σ-orthocomplete orthomodular poset (i.e. quantum logic) L and a probability measure (state) p defined on L. It follows from the very definition (see Sect. 6.1) that probability measures on quantum logics satisfy all numerical constraints imposed on Kolmogorovian probability measures: they are nonnegative, normalized, and σ-additive on families of

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“…The probabilities p(s) of the occurrence of an event obtained by observing the physical system for the different states s ∈ S determines a function from S to [0, 1], called a numerical event or multidimensional probability (cf. [1,2]). For example, one could think of finding the value of an observable within a given set of reals for different states s ∈ S.…”

Section: Introductionmentioning

confidence: 99%

“…p(s) ≤ q (s) for all s ∈ S, and call (p, q, r) an orthogonal triple if p ⊥ q ⊥ r ⊥ p. If one assumes that in addition to (1) and (2) it holds that (3) p, q, r ∈ P and (p, q, r) an orthogonal triple imply p + q + r ∈ P (with + the addition in R) then (P, ≤, ) is called an algebra of S-probabilities or algebra of multidimensional probabilities (cf. [1,2]). …”

Section: Introductionmentioning

confidence: 99%

On algebras of multidimensional probabilities

Dorfer

Dorninger

Länger

2010

Mathematica Slovaca

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ABSTRACT. The probability p(s) of the occurrence of an event pertaining to a physical system which is observed in different states s determines a function p from the set S of states of the system to [0,1]. The function p is called a multidimensional probability or numerical event. Sets of numerical events which are structured either by partially ordering the functions p and considering orthocomplementation or by introducing operations + and · in order to generalize the notion of Boolean rings representing classical event fields are studied with the goal to relate the algebraic operations + and · to the sum and product of real functions and thus to distinguish between classical and quantum mechanical behaviour of the physical system. Necessary and sufficient conditions for this are derived, as well for the case that the functions p can assume any value between 0 and 1 as for the special cases that the values of p are restricted to two or three different outcomes.

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On the characterization of probabilities: A generalization of Bell’s inequalities

Cited by 36 publications

References 4 publications

Boolean Properties and Bell-Like Inequalities of Numerical Events

Boolean Properties and Bell-Like Inequalities of Numerical Events

Perspectives

On algebras of multidimensional probabilities

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