A quantum system that interacts with a random environment is described by the density matrix of system plus environment, whose trace is taken with respect to the degrees of freedom of the environment. Actually, this sum is made because the environment is treated as a statistical ensemble. But in thermodynamics, quantities like internal energy that are averaged over the ensemble have very peaked distributions so that the mean is meaningful. But here there is only one environment at a time, and the off-diagonal elements of the density matrix have all the possible phases. Then the mean is 0, but that is meaningless.

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