Schroedinger bridges for classical and quantum discrete time Markovian evolutions
The theory of Schroedinger bridges for diffusion processes is extended to classical and quantum Markov chains. Taking into account the past-future lack of symmetry of the discrete-time setting, results bear a striking resemblance to the classical ones. In particular, the solution of the path space maximum entropy problems is always obtained from the a priori model by means of a suitable multiplicative functional transformation. In the quantum case, nonequilibrium time reversal of quantum channels is discussed and space-time harmonic processes are introduced.