Statistics of Measurement of Non-commuting Quantum Variables: Monitoring and Purification of a qubit

TL;DR

This paper develops a formalism for analyzing continuous weak measurements of a qubit's three noncommuting variables using three detectors, revealing high-fidelity monitoring and connections to quantum information algorithms.

Contribution

It introduces a formalism based on Feynman-Vernon and full counting statistics to evaluate measurement outcomes of noncommuting qubit variables.

Findings

High correlation between detector outcomes and qubit components.

Enables high-fidelity monitoring of all three pseudospin components.

Links measurement results to quantum information processing algorithms.

Abstract

We address continuous weak linear quantum measurement and argue that it is best understood in terms of statistics of the outcomes of the linear detectors measuring a quantum system, for example, a qubit. We mostly concentrate on a setup consisting of a qubit and three independent detectors that simultaneously monitor three noncommuting operator variables, those corresponding to three pseudospin components of the qubit. We address the joint probability distribution of the detector outcomes and the qubit variables. When analyzing the distribution in the limit of big values of the outcomes, we reveal a high degree of correspondence between the three outcomes and three components of the qubit pseudospin after the measurement. This enables a high-fidelity monitoring of all three components. We discuss the relation between the monitoring described and the algorithms of quantum information theory that use the results of the partial measurement. We develop a proper formalism to evaluate the statistics of continuous weak linear measurement. The formalism is based on Feynman-Vernon approach, roots in the theory of full counting statistics, and boils down to a Bloch-Redfield equation augmented with counting fields.