The latter is spanned in a direct product space of the physical space 𝑃𝑃 and an auxiliary space 𝑄𝑄 chosen as an exact copy of P. Then the thermal ensemble average 〈𝑂𝑂〉 at T can be expressed as the expectation value of pure thermal state �Ψ 𝛽𝛽 � (we suppose �Ψ 𝛽𝛽 � is normalized and 𝛽𝛽 = (𝑘𝑘 𝐵𝐵 𝑇𝑇) -1 ):