Efficient evaluation of fundamental sensitivity limits and full counting statistics for continuously monitored Gaussian quantum systems

TL;DR

This paper develops an efficient method to evaluate fundamental sensitivity limits and full counting statistics in Gaussian quantum systems under continuous monitoring, enabling precise quantum parameter estimation and thermodynamic analysis.

Contribution

It introduces a set of differential equations for Gaussian states that simplifies the analysis of monitored quantum systems, including fidelity, QFI, and counting statistics, without Hilbert-space truncation.

Findings

Efficient computation of fidelity and QFI for Gaussian systems.

Analytical solutions in simple monitoring scenarios.

Benchmarking of thermodynamic uncertainty relations.

Abstract

Generalized master equations (GMEs) -- time-local but generally neither trace-preserving nor Hermiticity-preserving -- are convenient tools to compute properties of the environment of an open or continuously monitored quantum system. A two-sided master equation yields the fidelity and quantum Fisher information (QFI) of environment states, thereby setting fundamental limits for hypothesis testing and parameter estimation under continuous monitoring. For unmonitored noise or inefficient detection, the QFI of the detectable part of the environment may be obtained from a recently derived GME acting on multiple system replicas. Tilted master equations provide the full counting statistics of quantum jumps and diffusive measurements, enabling, e.g., studies of quantum thermodynamics beyond average values. Here we focus on bosonic linear systems, governed by a quadratic Hamiltonian and linear jump operators, whose dynamics preserves Gaussianity. For Gaussian initial states, we recast a generic GME as a compact set of ordinary differential equations for the covariance matrix (a Riccati-type equation), first moments, and normalization. These equations can be integrated efficiently without Hilbert-space truncation, and admit analytical results in simple settings. We also provide specialized forms for fidelity/QFI and full counting statistics. We illustrate the formalism with a continuously monitored optical parametric oscillator, using it to determine sensitivity limits for frequency estimation and to benchmark Hasegawa's thermodynamic uncertainty relations.