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.

中文翻译：

---

热状态的最小误差判别

我们研究了热状态的最小误差判别问题的几种变体。除了提供错误概率的最佳值外，我们还表征了最佳测量值。对于固定哈密顿量的情况，我们表明对于一般判别问题，最佳测量是在哈密顿量的能量基中的测量。我们确定临界温度，确定给定温度是否与非常高或非常低温度的热状态最有区别。此外，我们研究了热状态是高于还是低于某个温度阈值的决策问题。此外，在这种情况下，最小误差测量是基于能量的测量。一旦要区分的热态具有不同的哈密顿量，情况就不再如此。我们分析了温度固定但哈密顿量不同的特定情况。对于所考虑的情况，我们表明最佳测量与固定温度和相互作用强度无关。