Thermometry of Gaussian quantum systems using Gaussian measurements

TL;DR

This paper investigates optimal Gaussian measurement strategies for estimating the temperature of Gaussian quantum systems, revealing regimes where specific measurements are optimal and discussing the potential advantages of joint measurements.

Contribution

It introduces a general numerical method to identify optimal Gaussian measurements for thermometry and provides analytical solutions for key cases, highlighting measurement regimes.

Findings

Heterodyne or Homodyne measurements are optimal depending on temperature.

Gaussian measurements are optimal at high temperatures.

Photo-detection-like measurements are optimal at low temperatures.

Abstract

We study the problem of estimating the temperature of Gaussian systems with feasible measurements, namely Gaussian and photo-detection-like measurements. For Gaussian measurements, we develop a general method to identify the optimal measurement numerically, and derive the analytical solutions in some relevant cases. For a class of single-mode states that includes thermal ones, the optimal Gaussian measurement is either Heterodyne or Homodyne, depending on the temperature regime. This is in contrast to the general setting, in which a projective measurement in the eigenbasis of the Hamiltonian is optimal regardless of temperature. In the general multi-mode case, and unlike the general unrestricted scenario where joint measurements are not helpful for thermometry (nor for any parameter estimation task), it is open whether joint Gaussian measurements provide an advantage over local ones. We conjecture that they are not useful for thermal systems, supported by partial analytical and numerical evidence. We further show that Gaussian measurements become optimal in the limit of large temperatures, while on/off photo-detection-like measurements do it for when the temperature tends to zero. Our results therefore pave the way for effective thermometry of Gaussian quantum systems using experimentally realizable measurements.