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arXiv:1409.3744

Bell-type inequalities for bivariate maps on orthomodular lattices

Jarosław Pykacz, L’ubica Valášková, Oľga Nánásiová

Published Sep 12, 2014• 0 citations
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Abstract

Bell-type inequalities on orthomodular lattices, in which conjunctions of propositions are not modeled by meets but by maps for simultaneous measurements (s-maps), are studied. It is shown that the most simple of these inequalities, that involves only two propositions, is always satisfied, contrary to what happens in the case of traditional version of this inequality in which conjunctions of propositions are modeled by meets. Equivalence of various Bell-type inequalities formulated with the aid of bivariate maps on orthomodular lattices is studied. Our invesigations shed new light on the interpretation of various multivariate maps defined on orthomodular lattices already studied in the literature. The paper is concluded by showing the possibility of using s-maps and j-maps to represent counterfactual conjunctions and disjunctions of non-compatible propositions about quantum systems.

DOI
10.48550/arxiv.1409.3744

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Bell-type inequalities on orthomodular lattices, in which conjunctions of propositions are not modeled by meets but by maps for simultaneous measurements (s-maps), are studied. It is shown that the most simple of these inequalities, that involves only two propositions, is always satisfied, contrary to what happens in the case of traditional version of this inequality in which conjunctions of propositions are modeled by meets. Equivalence of various Bell-type inequalities formulated with the aid of bivariate maps on orthomodular lattices is studied. Our invesigations shed new light on the interpretation of various multivariate maps defined on orthomodular lattices already studied in the literature. The paper is concluded by showing the possibility of using s-maps and j-maps to represent counterfactual conjunctions and disjunctions of non-compatible propositions about quantum systems.
Bell-type inequalities on orthomodular lattices, in which conjunctions of propositions are not modeled by meets but by maps for simultaneous measurements ( sss -maps), are studied. It is shown, that the most simple of these inequalities, that involves only two propositions, is always satisfied, contrary to what happens in the case of traditional version of this inequality in which conjunctions of propositions are modeled by meets. Equivalence of various Bell-type inequalities formulated with the aid of bivariate maps on orthomodular lattices is studied. Our investigations shed new light on the interpretation of various multivariate maps defined on orthomodular lattices already studied in the literature. The paper is concluded by showing the possibility of using sss -maps and jjj -maps to represent counterfactual conjunctions and disjunctions of non-compatible propositions about quantum systems.
Mathematical description of quantum phenomena requires an event structure more general than a Boolean algebra. For this purpose, Birkhoff and von Neumann proposed the notion of an orthomodular lattice. Its typical feature is the existence of so–called non–commuting events, which are not simultaneously observable (like position and momentum, according to Heisenberg’s uncertainty principle). There is an old open problem whether the word problem for orthomodular lattices is solvable. Is it possible to decide whether two formulas are equivalent? In Boolean algebras, an easy positive answer is given by a transformation of the formula to a unique normal form. This requires the commutativity, associativity, and distributivity of the Boolean operations (disjunction and conjunction). In orthomodular lattices, the corresponding lattice operations (join and meet) violate distributivity. This disables the use of normal forms. We looked for alternative approaches. E.g., the join is only one of six orthomodular lattice operations generalizing the disjunction. In the thesis, we study the question whether some of the 96 binary operations in orthomodular lattices admit normal forms similar to the classical logic. The first question was which operations satisfy the associative identity, eventually under the assumption that some variables commute. Then we studied monotonicity because it is related to distributivity over the meet and join. The conclusion is that there is no pair of operations in orthomodular lattices admitting “Boolean–like” normal forms. In the last chapter we study “Moufang–like” identities, which were inspired by the algebras of quaternions and octonions. These identities generalize associativity and may enable further progress. As a by–product, we proved interesting, yet unknown, properties of some orthomodular lattice operations (e.g., the Sasaki projection). These new tools simplify algebraic computations and give a chance to develop algorithms more general than the current specialized software.
The theory of orthomodular lattice and its relationships that depend on s-map and condi-tional states definitions has shown in several types of probabilistic relations. An orthomodularlattice with a conditional states can be applied and defined as a model for non-compatible events.Studying of s-maps or conditional states on an orthomodular lattice helps us to describe suchproperties of random events, which are difficult to be described by Boolean algebra. In a par-ticularly observed quantum or classical system the information, that we cannot access results inour description of the system be coming mixed even if we have perfect initial knowledge.That is, if the system is quantum the conditional states will be given by a state matrix and ifclassical, the conditional states will be given by a probability distribution. We show conditionswhen a quantum system has the same properties as the classical probability space.Acknowledgement. This work was supported by Science and Technology Assistance AgencyunderthecontractNo. APVT-20-003204, VEGA-1/3014/06, VEGA-1/3321/06, VEGA-1/3006/06
We present an analysis of the structure of Bell inequalities, mainly for the case of N qubits with two observables each. We show that these inequalities are related to Hadamard matrices and define Bell polynomials (in one variable) as an additional tool. With these aids we raise several conditions the coefficients of Bell inequalities must satisfy, and recursively generate the whole set of inequalities starting from N=1. Moreover, we prove some characteristic features of this set, such as that most of the inequalities contain all expectation values under consideration. Finally, we show how the presented results can be used to construct Bell inequalities with certain properties. An outlook on further research topics concludes the paper.
We remind the viewpoint that violation of Bell's inequality might be interpreted not only as an evidence of the alternative -- either nonlocality or ``death of reality'' (under the assumption the quantum mechanics is incomplete). Violation of Bell's type inequalities is a well known sufficient condition of incompatibility of random variables -- impossibility to realize them on a single probability space. Thus, in fact, we should take into account an additional interpretation of violation of Bell's inequality -- a few pairs of random variables (two dimensional vector variables) involved in the EPR-Bohm experiment are incompatible. They could not be realized on a single Kolmogorov probability space. Thus one can choose between: a) completeness of quantum mechanics; b) nonlocality; c) `` death of reality''; d) non-Kolmogorovness. In any event, violation of Bell's inequality has a variety of possible interpretations. Hence, it could not be used to derive the unique conclusion on the relation between quantum and classical models.