  We consider quantum circuits with time travel designed for distinguishing
specific nonorthogonal quantum states in two most popular models: Deutschs and
postselected. We modify them by a presence of weakly coupled thermal
environment. Using the Davies approximation we study how the thermal noise
affects an ability of the circuits to distinguish nonorthogonal quantum states.
We show that for purely dephasing environment a paradoxial power of such
circuits remains preserved. We also present a physics based argument for
conditions of validity of the maximum entropy rule introduced by David Deutsch
for resolving the uniqueness ambiguity in a circuit with time travel.
