Minimum-error discrimination of thermal states

TL;DR

This paper investigates the optimal strategies for distinguishing thermal quantum states with minimum error, identifying conditions under which energy basis measurements are optimal and analyzing the impact of temperature and Hamiltonian differences.

Contribution

It provides the optimal measurement strategies for thermal state discrimination, including the role of the energy basis and effects of temperature and Hamiltonian variations.

Findings

Optimal measurement is the energy basis for fixed Hamiltonian discrimination.

Identifies a critical temperature influencing distinguishability.

Shows measurement independence from temperature and interaction strength when Hamiltonians differ.

Abstract

We study several variations of the question of minimum-error discrimination of thermal states. Besides of providing the optimal values for the probability of error, we also characterize the optimal measurements. For the case of a fixed Hamiltonian, we show that for a general discrimination problem the optimal measurement is the measurement in the energy basis of the Hamiltonian. We identify a critical temperature, determining whether the given temperature is best distinguishable from thermal state of very high or very low temperatures. Further, we investigate the decision problem of whether the thermal state is above or below some threshold value of the temperature. Also, in this case, the minimum-error measurement is the measurement in the energy basis. This is no longer the case once the thermal states to be discriminated have different Hamiltonians. We analyze a specific situation when the temperature is fixed but the Hamiltonians are different. For the considered case, we show the optimal measurement is independent of the fixed temperature and also of the strength of the interaction.