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

Francesco Albarelli, Marco G. Genoni

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.

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

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.
Paper Structure (8 sections, 65 equations, 3 figures)

This paper contains 8 sections, 65 equations, 3 figures.

Table of Contents

  1. Trace norm of a Gaussian operator
  2. Gaussian-integral evaluation of the moments of $K=\hat{\nu}^\dagger \hat{\nu}$
  3. Closed formulas
  4. (i) Normalization.
  5. (ii) Covariance matrix.
  6. Details on the OPO calculations
  7. Derivation of frequency estimation bounds
  8. Derivation of full counting statistics and Hasegawa's TUR bound

Figures (3)

  • Figure 1: Fisher information rates for $\omega$ estimation with an optical parametric oscillator (OPO), as a function of time. The signal CFI and unravelling QFI are obtained for continuous homodyne detection.
  • Figure 2: The ratio $\mathtt{D}/\mathtt{J}^2$ and the right-hand side $1/f$ of Hasegawa's TUR for the OPO, as a function of $\chi / \kappa$, with $\omega = 0$.
  • Figure 3: Fisher information rates for $\omega$ estimation with an inefficiently ($\eta = 0.5$) monitored optical parametric oscillator (OPO), as a function of time.