Stochastic sequential reduction of commutative Hamiltonians

Abstract

This paper introduces a class of stochastic processes constructed by conditioning cadlag 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 cadlag 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 Schrodinger 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.