Operational Markov Condition for Quantum Processes

We derive a necessary and sufficient condition for a quantum process to be Markovian which coincides with the classical one in the relevant limit. Our condition unifies all previously known definitions for quantum Markov processes by accounting for all potentially detectable memory effects. We then derive a family of measures of non-Markovianity with clear operational interpretations, such as the size of the memory required to simulate a process or the experimental falsifiability of a Markovian hypothesis.

Figure 1

Determining whether a quantum process is Markovian. Generalized operations ${\mathbf{A}}_{k:0}$ are made on the system during a quantum process, where the subscripts represent the time. At time step $k$, we make a causal break by measuring the system with ${\mathrm{\Pi }}_k^{(r)}$ and repreparing it in randomly chosen state $P_k^{(s)}$. The process is said to be Markovian if and only if ${\rho }_l(P_k|{\mathrm{\Pi }}_k^{(r)};{\mathbf{A}}_{k-1:0})={\rho }_l(P_k^{(s)})$ at all time steps $l$, $k$, for all inputs $P_k^{(s)}$, measurements $\{{\mathrm{\Pi }}_k^{(r)}\}$, and control operations $\{{\mathbf{A}}_{k-1:0}\}$.