Statistics of Measurement of Non-commuting Quantum Variables

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 develop a proper formalism to evaluate the statistics of such measurement. Generally, we are able to evaluate the joint probability distribution of the detector outcomes and the qubit variables. We concentrate on two setups. The application of our method to the setup where a single pseudospin component is measured gives a comphrehensive picture of quantum non‐demolition measurement. More interesting setup consists of a qubit and three independent detectors that simultaneously monitor three non‐commuting operator variables, those corresponding to three pseudo‐spin components of the qubit. 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 pseudo‐spin 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. 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.