Weak Values with Decoherence

TL;DR

This paper introduces a new framework for understanding weak values in quantum mechanics, incorporating decoherence effects and analyzing their impact on geometric phases under noise.

Contribution

It presents a novel weak operator formalism and a general quantum operation framework for weak values, including decoherence effects and their influence on geometric phases.

Findings

Decoherence causes a shift in the probe wave function during weak measurements.

The geometric phase is affected by bit flip noise in the presence of decoherence.

A new formalism for weak operators is proposed, extending the Kraus representation.

Abstract

The weak value of an observable is experimentally accessible by weak measurements as theoretically analyzed by Aharonov et al. and recently experimentally demonstrated. We introduce a weak operator associated with the weak values and give a general framework of quantum operations to the W operator in parallel with the Kraus representation of the completely positive map for the density operator. The decoherence effect is also investigated in terms of the weak measurement by a shift of a probe wave function of continuous variable. As an application, we demonstrate how the geometric phase is affected by the bit flip noise.