Complex counterpart of variance in quantum measurements for pre- and post-selected systems

TL;DR

This paper introduces the concept of weak variance, a complex-valued extension of variance for pre- and post-selected quantum systems, and demonstrates its physical effects and statistical interpretation through optical experiments.

Contribution

It formulates the weak variance as a complex quantity, linking its real and imaginary parts to observable changes in probe wave packets and providing a statistical framework.

Findings

Experimental demonstration of wave packet width changes due to weak variance

Weak variance can be expressed as the variance of a weak-valued probability distribution

Supports the interpretation of weak variance as a complex extension of quantum variance

Abstract

The variance of an observable in a pre-selected quantum system, which is always real and non-negative, appears as an increase in the probe wave packet width in indirect measurements. Extending this framework to pre- and post-selected systems, we formulate a complex-valued counterpart of the variance called "weak variance." In our formulation, the real and imaginary parts of the weak variance appear as changes in the probe wave packet width in the vertical-horizontal and diagonal-antidiagonal directions, respectively, on the quadrature phase plane. Using an optical system, we experimentally demonstrate these changes in the probe wave packet width caused by the real negative and purely imaginary weak variances. Furthermore, we show that the weak variance can be expressed as the variance of the weak-valued probability distribution in pre- and post-selected systems. These operational and statistical interpretations support the rationality of formulating the weak variance as a complex counterpart of the variance in pre- and post-selected systems.