Discrete-time classical and quantum Markovian evolutions: Maximum   entropy problems on path space
  The theory of Schroedinger bridges for diffusion processes is extended to classical and quantum discrete-time Markovian evolutions. The solution of the path space maximum entropy problems is obtained from the a priori model in both cases via a suitable multiplicative functional transformation. In the quantum case, nonequilibrium time reversal of quantum channels is discussed and space-time harmonic processes are introduced. 
Michele Pavon and Francesco Ticozzi