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 independently of the model of the microscopic interaction and is valid for both classical and quantum stochastic processes, relies on a quantum version of the so-called Blackwell-Sherman-Stein theorem in classical statistics.

Figure 1

Schematic representation of a discrete-time dynamical mapping. Top: The system's evolution from an initial time $t_0$ to the final time $t_N$ is represented by a sequence of quantum channels ${\left({\mathcal{N}}^i\right)}_{0\le i\le N}$ (the thin arrows), describing the stochastic evolution of the system from the initial time $t_0$ to some later time $t_i\ge t_0$. Bottom: The dynamical mapping is divisible if there exists another sequence of quantum channels ${\left({\mathcal{C}}^i\right)}_{1\le i\le N}$ (the thick arrows) such that ${\mathcal{N}}^{i+1}={\mathcal{C}}^{i+1}\circ {\mathcal{N}}^i$ for all $0\le i\le N-1$.

Figure 2

Top: The joint system-environment evolution. Middle: The reduced dynamics of the system is divisible if it cannot be distinguished from a sequence of independent quantum channels ${\left({\mathcal{C}}^i\right)}_{i\ge 1}$ applied in series. Bottom: As a consequence of the Stinespring-Kraus representation theorem [29, 30], any divisible process can always be thought of as originating from a collisionlike model, in which the system interacts with an environment that is reset after every interaction.