Quantum Markov Order

We formally extend the notion of Markov order to open quantum processes by accounting for the instruments used to probe the system of interest at different times. Our description recovers the classical property in the appropriate limit: when the stochastic process is classical and the instruments are noninvasive, i.e., restricted to orthogonal, projective measurements. We then prove that there do not exist non-Markovian quantum processes that have finite Markov order with respect to all possible instruments; the same process exhibits distinct memory effects when probed by different instruments. This naturally leads to a relaxed definition of quantum Markov order with respect to specified instrument sequences. The memory effects captured by different choices of instruments vary dramatically, providing a rich landscape for future exploration.

Figure 1

Knowledge of the $\ell =2$ states in the memory block suffices to predict the future statistics via the recovery map ${\mathcal{W}}_{M\to FM}$. No information about the history is required to determine these probabilities, indicated by the question mark. This property holds true for all time steps. Any influence the history (beyond $\ell$ time steps ago) has on the future is mediated through memory blocks; conditional on the states in the most recent block, there are no correlations between history and future, indicated by the dashed arrow.

Figure 2

An instrument ${\mathcal{J}}_M$, comprising (temporally correlated) CP maps $\{O_M^{(x_M)}\}$ (green) over $\ell$ time steps, is applied to a process ${\mathrm{\Upsilon }}_{FMH}$. The process has Markov order $\ell$ with respect to this instrument, denoted ${\ell }^{{\mathcal{J}}_M}$ when, for each possible realization $x_M$, the history (red) and future (blue) conditional processes are independent.