Statistics of continuous weak quantum measurement of an arbitrary quantum system with multiple detectors

TL;DR

This paper develops a comprehensive theoretical framework for analyzing continuous weak quantum measurements of arbitrary systems with multiple detectors, unifying various approaches and ensuring consistent interpretation of measurement results.

Contribution

It introduces a general, microscopic, and phenomenological framework for continuous quantum measurements involving multiple detectors, unifying different methods and establishing conditions for valid measurement interpretation.

Findings

Unified various approaches to continuous quantum measurement

Derived equations from microscopic and phenomenological models

Established conditions for measurement result interpretation

Abstract

In this paper, we establish a general theoretical framework for the description of continuous quantum measurements and the statistics of the results of such measurements. The framework concerns the measurement of an arbitrary quantum system with arbitrary number of detectors under realistic assumption of instant detector reactions and white noise sources. We attend various approaches to the problem showing their equivalence. The approaches include the full counting statistics (FCS) evolution equation a for pseudo-density matrix, the drift-diffusion equation for a density matrix in the space of integrated outputs, and discrete stochastic updates. We provide the derivation of the underlying equations from a microscopic approach based on full counting statistics method, a phenomenological approach based on Lindblad construction, and interaction with auxiliary quantum systems representing the detectors. We establish the necessary conditions on the phenomenological susceptibilities and noises that guarantee the unambiguous interpretation of the measurement results and the positivity of the density matrix. Our results can be easily extended to describe various quantum feedback schemes where the manipulation decision is based on the values of detector outputs.