Assessing Non-Markovian Quantum Dynamics

We investigate what a snapshot of a quantum evolution—a quantum channel reflecting open system dynamics—reveals about the underlying continuous time evolution. Remarkably, from such a snapshot, and without imposing additional assumptions, it can be decided whether or not a channel is consistent with a time (in)dependent Markovian evolution, for which we provide computable necessary and sufficient criteria. Based on these, a computable measure of “Markovianity” is introduced. We discuss how the consistency with Markovian dynamics can be checked in quantum process tomography. The results also clarify the geometry of the set of quantum channels with respect to being solutions of time (in)dependent master equations.

Figure 1

Schematic depiction of the (12-dimensional) convex set of qubit channels. The (dark gray) subset of Markovian channels is nonconvex and contains 2% of the channels. The larger nonconvex set of time-dependent Markovian channels (17%) contains all extremal channels. All sets, including the measure zero set of indivisible channels (black line), can be found in the neighborhood of the identity (dotted circle).

Figure 2

Deviation from Markovianity. (a) For a mixture of a $\pi /4-{\sigma }_x$ rotation ($p=1$) and a dephasing channel ($p=0$). (b) For the damped Jaynes-Cummings model (as a function of time), in which a single spin or qubit is coupled to a single cavity mode undergoing lossy dynamics. The field mode serves as an intermediate system preserving correlations that are relevant for the systems’s dynamics. The figure shows the interplay between the time scale of truly irreversible cavity losses and apparent decay on the time scale of oscillations, leaving intervals which are consistent with a Markovian process ($\omega =0.2$, $\gamma =0.35$, ${\alpha }_{x,z}=1/2$, ${\alpha }_y=1$). Also shown (dotted line) is the evolution of $⟨0|\rho |0⟩$ for initial condition $⟨1|\rho |1⟩=1$.