Classical Analogue of Weak Value in Stochastic Process

TL;DR

This paper demonstrates a classical analogue of the quantum weak value using a stochastic process, clarifying its unusual behaviors without complex probabilities.

Contribution

It introduces a classical stochastic model that emulates quantum weak values through a two-time conditional expectation, providing new insights into their nature.

Findings

Weak values can be represented as two-time conditional expectations in classical models.

Negative probabilities and expectation enhancements are explained without complex numbers.

The symmetrized master equation parallels quantum wave equations, illuminating weak value phenomena.

Abstract

One of the remarkable notions in the recent development of quantum physics is the weak value related to weak measurements. We emulate it as a two-time conditional expectation in a classical stochastic model. We use the well known symmetrized form of the master equation, which is formally equivalent to the wave equation in quantum mechanics apart from the fact that wave functions are always real. The origin of the unusual behaviors of the weak value such as the negative probability and the abnormal enhancement of some expectations becomes clearer in the present case, where the two-time conditional probability has no ambiguity of imaginary/complex values.