  Detailed fluctuation theorems are statements about the probability
distribution for the stochastic entropy production along a trajectory. It
involves the consideration of a suitably transformed dynamics, such as the time
reversed, the adjoint, or a combination of these. We identify specific,
typically unique, initial conditions, called echo states, for which the final
probability distribution of the transformed dynamics reproduces the initial
distribution. In this case the detailed fluctuation theorems relate the
stochastic entropy production of the direct process to that of the transformed
one. We illustrate our results by an explicit analytical calculation and
numerical simulations for a modulated two-state quantum dot.
