Weak value expansion of quantum operators and its application in stochastic matrices

TL;DR

This paper introduces a weak value expansion for Hermitian operators using biorthogonal basis, providing new insights into weak values and applying this to identify stochastic matrices in quantum measurements.

Contribution

It presents a novel weak value expansion of Hermitian operators and demonstrates its application in analyzing stochastic matrices in quantum measurement scenarios.

Findings

Hermitian operators can be expanded using biorthogonal basis and weak values.

The approach reveals irreversible stochastic matrices in quantum measurements.

Provides new physical interpretation of weak values.

Abstract

It is shown that any Hermitian operator can be expanded in terms of a set of operators formed from biorthogonal basis, and the expansion coefficients are given as products of weight functions and weak values, shedding a new light on the physical interpretation of the weak value. The utility of our approach is showcased with examples of spin one-half and spin one systems, where irreversible subset of stochastic matrices describing projective measurement on a mixed state is identified.