  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.
