Exact quantum Bayesian rule for qubit measurements in circuit QED

TL;DR

This paper derives an exact, analytical quantum Bayesian rule for qubit measurements in circuit QED, improving efficiency and accuracy over numerical methods and enabling advanced quantum state analysis.

Contribution

It provides a rigorous, generalized quantum Bayesian approach for circuit QED measurements, enhancing existing methods with an exact analytical solution.

Findings

Improves accuracy of qubit state updates during measurement.

Enables analytical studies of quantum weak values and state smoothing.

Offers a more efficient alternative to numerical QTE integration.

Abstract

Developing efficient framework for quantum measurements is of essential importance to quantum science and technology. In this work, for the important superconducting circuit-QED setup, we present a rigorous and analytic solution for the effective quantum trajectory equation (QTE) after polaron transformation and converted to the form of Stratonovich calculus. We find that the solution is a generalization of the elegant quantum Bayesian approach developed in arXiv:1111.4016 by Korotokov and currently applied to circuit-QED measurements. The new result improves both the diagonal and offdiagonal elements of the qubit density matrix, via amending the distribution probabilities of the output currents and several important phase factors. Compared to numerical integration of the QTE, the resultant quantum Bayesian rule promises higher efficiency to update the measured state, and allows more efficient and analytical studies for some interesting problems such as quantum weak values, past quantum state, and quantum state smoothing. The method of this work opens also a new way to obtain quantum Bayesian formulas for other systems and in more complicated cases.