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arXiv:quant-ph/0312183

Marginality in non-compatible random events

Oľga Nánásiová, Andrei Yu. Khrennikov

Published Dec 22, 2003• 0 citations
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Abstract

We present a way of introducing joint distibution function and its marginal distribution functions for non-compatible observables. Each such marginal distribution function has the property of commutativity. Models based on this approach can be used to better explain some classical phenomena in stochastic processes.
DOI
10.48550/arxiv.quant-ph/0312183

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This paper introduces a class of stochastic processes constructed by conditioning càdlàg processes to take a predetermined set of marginal laws at fixed points in time. We collectively refer to the elements of this class of processes as random n-bridges (RnBs), where n refers to the cardinality of the set of conditioning laws, which can be chosen arbitrarily. We prove that if the underlying càdlàg process is Markov, then its RnB is Markov for n = 1 and has a particular Markov-like property for n > 1. As a canonical subclass, we construct Brownian RnBs (BRnBs) and provide their anticipative representation as well as their non-anticipative semimartingale representation. This motivates the second part of this paper, wherein we apply BRnBs to describe an energy-based state reduction in a composite quantum system involving a set of commuting observables measured successively at different times. The highly tractable so-called energy-based quantum state reduction models are significant in their role as the most important alternative to the popular CLS model. We ask the random variables composing the anticipative form of a BRnB to take the eigenvalues of a set of compatible Hamiltonians and provide a new stochastic Schrödinger evolution on a complex Hilbert space that models sequential reduction dynamics of the eigenstates through a single wave function. By doing so, we also extend the Brody–Hughston finite-time collapse model to a many-body setup with sequential measurements.
An empirical model is a generalization of a probability space. It consists of a simplicial complex of subsets of a class 𝒳 of random variables such that each simplex has an associated probability distribution. The ensuing marginalizations are coherent, in the sense that the distribution on a face of a simplex coincides with the marginal of the distribution over the entire simplex. An empirical model is called contextual if its distributions cannot be obtained by marginalizing a joint distribution over 𝒳. Contextual empirical models arise naturally in quantum theory, giving rise to some of its counter -intuitive statistical consequences. In this paper, we present a different and classical source of contextual empirical models: the interaction among many stochastic processes. We attach an empirical model to the ensuing network in which each node represents an open stochastic process with input and output random variables. The statistical behaviour of the network in the long run makes the empirical model generically contextual and even strongly contextual.
Suppose that a family of commutative measurement contexts is given, together with the corresponding classical probability laws. We study the problem under what conditions these contextwise descriptions can be combined into a single classical representation. We distinguish two ways in which this can be done. In {\em global integration}, the prescribed observable algebras are required to fit into a single commutative context and the corresponding probability laws to arise as marginals of one joint law. In {\em protocolwise integration}, by contrast, the context label is retained as part of the description, and the contextwise laws appear as conditional distributions within a single model of the measurement protocol. We show that global integration involves two distinct notions of compatibility: operator-algebraic compatibility of the underlying observables and probabilistic compatibility of their classicalised laws. The former implies the latter, but the converse fails; under the topological hypotheses stated below, probabilistic compatibility is characterised by a Kellerer-type marginal criterion. Protocolwise integration, on the other hand, always exists and is represented operator-algebraically by a direct-sum construction. When global integration exists, the two constructions arise as reductions of a common commutative refinement. We illustrate the distinction in the Bell--CHSH experiment, where Fine's theorem relates global probabilistic integration to Bell locality and the CHSH inequalities. This shows that the obstruction revealed by Bell--CHSH concerns not classical representation as such, but classical representation preserving specified cross-context identifications.