Equivalence between divisibility and monotonic decrease of information   in classical and quantum stochastic processes
  The crucial feature of a memoryless stochastic process is that any information about its state can only decrease as the system evolves. Here we show that such a decrease of information is equivalent to the underlying stochastic evolution being divisible. The main result, which holds for both classical and quantum stochastic processes, rely on a quantum version of the so-called Blackwell-Sherman-Stein theorem in classical statistics. 
Francesco Buscemi, Nilanjana Datta
quant-ph cond-mat.stat-mech