Quantum stochastic processes

Farrukh Maksutovich Mukhamedov

In [1] the non-stationaryMarkov process generated by a quantum quadratic stochastic process (QQSP) was defined, and it was noted that if the QQSP satisfies the ergodic principle, then so does the Markov process. In the present article we use a decomposition of a QQSP (see [2]) to prove that the converse is also true, that is, if the Markov process satisfies the ergodic principle, then so does the corresponding QQSP.

E. B. Davies

R. L. Hudson

Simon Milz, Kavan Kishore Modi

The field of classical stochastic processes forms a major branch of mathematics.Stochastic processes are, of course, also very well studied in biology, chemistry, ecology, geology, finance, physics, and many more fields of natural and social sciences.When it comes to quantum stochastic processes, however, the topic is plagued with pathological issues that have led to fierce debates amongst researchers.Recent developments have begun to untangle these issues and paved the way for generalizing the theory of classical stochastic processes to the quantum domain without ambiguities.This tutorial details the structure of quantum stochastic processes, in terms of the modern language of quantum combs, and is aimed at students in quantum physics and quantum-information theory.We begin with the basics of classical stochastic processes and generalize the same ideas to the quantum domain.Along the way, we discuss the subtle structure of quantum physics that has led to troubles in forming an overarching theory for quantum stochastic processes.We close the tutorial by laying out many exciting problems that lie ahead in this branch of science.

Anatolij Dvurečenskij, Sylvia Pulmannová

Ying Oon Tan

Previously [2004, AIP Conf. Proc., 750, 361], we considered a stochastic mechanics in the form of a stochastic differential equation mdxdt = μ(t) where μ(t) is a stochastic process defined by the set of Bohmian momentum time histories from an ensemble of particles. In this paper, we consider another stochastic differential equation d2xdt2 = 1m[−dV(x)dx + ζ(t)] where the stochastic force ζ(t) is a stochastic process defined by the set of quantum force time histories from an ensemble of particles. We show that, if the stochastic process is a purely random process characterized by an n‐th order joint probability density in the form of products of delta functions, then the stochastic mechanics with the stochastic force ζ(t) is equivalent to quantum mechanics in the sense that the former yields the same position probability density as the latter. However, for a particular non‐purely random process, we show that the stochastic mechanics with the stochastic force ζ(t) is not equivalent to quantum mechanics. Therefore, this stochastic mechanics is also generally not equivalent to quantum mechanics.