Weak values in a classical theory with an epistemic restriction

TL;DR

This paper demonstrates that weak values, typically associated with quantum mechanics, can be understood within a classical framework with an epistemic restriction, providing an operational interpretation and insights into measurement disturbance.

Contribution

It shows that weak values can be derived and interpreted in a classical theory with an epistemic restriction, bridging quantum and classical perspectives.

Findings

Real and imaginary parts of weak values correspond to phase space displacements.

Imaginary part measures postselection bias in phase space distributions.

Biases vanish as measurement disturbance approaches zero.

Abstract

Weak measurement of a quantum system followed by postselection based on a subsequent strong measurement gives rise to a quantity called the weak value: a complex number for which the interpretation has long been debated. We analyse the procedure of weak measurement and postselection, and the interpretation of the associated weak value, using a theory of classical mechanics supplemented by an epistemic restriction that is known to be operationally equivalent to a subtheory of quantum mechanics. Both the real and imaginary components of the weak value appear as phase space displacements in the postselected expectation values of the measurement device's position and momentum distributions, and we recover the same displacements as in the quantum case by studying the corresponding evolution in the classical theory. By using this analogous classical theory, we gain insight into the appearance of the weak value as a result of the statistical effects of post selection, and this provides us with an operational interpretation of the weak value, both its real and imaginary parts. We find that the imaginary part of the weak value is a measure of how much postselection biases the mean phase space distribution for a given amount of measurement disturbance. All such biases proportional to the imaginary part of the weak value vanish in the limit where disturbance due to measurement goes to zero. Our analysis also offers intuitive insight into how measurement disturbance can be minimised and the limits of weak measurement.