Ju l 2 02 1 Discrimination of thermal states

We study several variations of the question of minimum-error discrimination of thermal states. Except of providing the optimal values for the probability error we also characterize the optimal measurements. For the case of fixed Hamiltonian we show that for a general discrimination problem the optimal measurement is the measurement in the energy basis of the Hamiltonian. We identified 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 whether the thermal state is above, or below some threshold value of the temperature. Also in this case the optimal 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 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.