The information about the state of a qubit gained by a weakly coupled detector

TL;DR

This paper investigates how information about a qubit’s state can be obtained through weak measurement by a nearby detector, even when the qubit’s Hamiltonian and measured operator do not commute, revealing stochastic measurement bases and robust information rates.

Contribution

It introduces a detailed analysis of information extraction in weak qubit measurements with non-commuting observables, including stochastic measurement basis determination and an extended stochastic master equation.

Findings

Useful information can be extracted despite non-commuting measurements.

The measurement basis is stochastically determined each run.

Information acquisition rate remains largely unaffected by qubit dynamics.

Abstract

We analyze the information that one can learn about the state of a quantum two-level system, i.e. a qubit, when probed weakly by a nearby detector. In particular, we focus on the case when the qubit Hamiltonian and the qubit's operator being probed by the detector do not commute. Because the qubit's state keeps evolving while being probed and because the measurement data is mixed with a detector-related background noise, one might expect the detector to fail in this case. We show, however, that under suitable conditions and by proper analysis of the measurement data useful information about the state of the qubit can be extracted. It turns out that the measurement basis is stochastically determined every time the experiment is repeated. We analyze in detail the probability distributions that govern the choice of measurement bases. We also analyze the information acquisition rate and show that it is largely unaffected by the apparent conflict between the measurement and intrinsic qubit dynamics. We discuss the relation between our analysis and the stochastic master equation that describes the evolution of the qubit's state under the influence of measurement and decoherence. In particular, we write down a stochastic equation that encompasses the usual stochastic master equation for the evolution of the qubit's density matrix and additionally contains the measurement information that can be extracted from the observed signal.