Bayesian quantum thermometry based on thermodynamic length

TL;DR

This paper introduces a Bayesian quantum thermometry framework utilizing thermodynamic length, addressing prior uncertainty and limited data, with applications to spin-1/2 systems probing heat reservoirs or samples.

Contribution

It develops a novel Bayesian approach to quantum temperature estimation based on thermodynamic length, suitable for scenarios with prior uncertainty and limited measurements.

Findings

Differentiates between reservoir probing and sample probing scenarios.

Demonstrates the effectiveness of the thermodynamic length approach.

Provides insights into fundamental limits of quantum thermometry.

Abstract

In this work, we propose a theory of temperature estimation of quantum systems, which is applicable in the regime of non-negligible prior temperature uncertainty and limited measurement data. In this regime the problem of establishing a well-defined measure of estimation precision becomes non-trivial, and furthermore the construction of a suitable criterion for optimal measurement design must be re-examined to account for the prior uncertainty. We propose a fully Bayesian approach to temperature estimation based on the concept of thermodynamic length, which solves both these problems. As an illustration of this framework, we consider thermal spin-$1/2$ particles and investigate the fundamental difference between two cases; on the one hand, when the spins are probing the temperature of a heat reservoir and, on the other, when the spins themselves constitute the sample.