Weak values in continuous weak measurement of qubits

TL;DR

This paper derives exact expressions for weak values in continuous weak qubit measurements, revealing non-perturbative effects and practical measurement techniques, especially in circuit-QED systems, enhancing quantum state tomography.

Contribution

It provides new exact formulas for weak values in continuous measurements, including finite strength and non-perturbative corrections, improving understanding and measurement methods in quantum systems.

Findings

Exact weak value expressions from quantum trajectory and Bayesian approaches.

Non-perturbative corrections are significant even in very weak limits.

Homodyne measurement phase tuning enables efficient quantum state tomography.

Abstract

For continuous weak measurement of qubits, we obtain exact expressions for weak values (WVs) from the post-selection restricted average of measurement outputs, by using both the quantumtrajectory- equation (QTE) and quantum Bayesian approach. The former is applicable to short-time weak measurement, while the latter can relax the measurement strength to finite. We find that even in the "very" weak limit the result can be essentially different from the one originally proposed by Aharonov, Albert and Vaidman (AAV), in a sense that our result incorporates non-perturbative correction which could be important when the AAV's WV is large. Within the Bayesian framework, we obtain also elegant expressions for finite measurement strength and find that the amplifier's noise in quantum measurement has no effect on the WVs. In particular, we obtain very useful result for homodyne measurement in circuit-QED system, which allows for measuring the real and imaginary parts of the AAV's WV by simply tuning the phase of the local oscillator. This advantage can be exploited as efficient state-tomography technique.