Completely Positive Divisibility Does Not Mean Markovianity

In the classical domain, it is well known that divisibility does not imply that a stochastic process is Markovian. However, for quantum processes, divisibility is often considered to be synonymous with Markovianity. We show that completely positive divisible quantum processes can still involve non-Markovian temporal correlations, that we then fully classify using the recently developed process tensor formalism, which generalizes the theory of stochastic processes to the quantum domain.

Figure 1

Circuits for checking ICP and OCP divisibility. (a) To construct the maps ${\mathrm{\Lambda }}_{s:0}$ and ${\mathrm{\Lambda }}_{t:0}$ we may set $r=0$ and measure the system at $s$ or $t$ respectively. (b) To construct ${\mathrm{\Lambda }}_{t:s}$, the system is discarded at $s_{-}$ and a fresh state is fed in at $s_{+}$. The dotted line encapsulates the three-time process tensor $T_{t:s:r}$.

Figure 2

Divisibility and Markovianity. (a) The system interacts with one part of the correlated environment state leading to a non-Markovian OCP-divisible process. (b) The hierarchy of sets of processes with varying degrees of temporal correlations.