Markovian and non-Markovian dynamics induced by a generic environment

We study the open dynamics of a quantum two-level system coupled to an environment modeled by random matrices. Using the quantum channel formalism, we investigate different quantum Markovianity measures and criteria. A thorough analysis of the whole parameter space reveals a wide range of different regimes, ranging from strongly non-Markovian to Markovian dynamics. In contrast to analytical models, all non-Markovianity measures and criteria have to be applied to data with fluctuations and statistical uncertainties. We discuss the practical usefulness of the different approaches.

Figure 1

Heat map of the process ending time $t_{\mathrm{End}}$ as a function of the qubit's energy splitting $\mathrm{\Delta }$ and the strength of coupling with its environment $\lambda$.

Figure 2

BLP measure (upper panel) and RHP measure (lower panel) for non-Markovian dynamics as a function of $\mathrm{\Delta }$ and $\lambda$. The parameter region considered is the same as in Fig. 1; again we use a log scale in the heat map representing the values of the NM measures. Both measures, defined in Eqs. (25) and (13), respectively, are taken at $t=t_{\mathrm{End}}$.

Figure 3

Maximum distinguishability recovery (MDR). The quantity ${\mathcal{N}}_{\mathrm{MDR}}$ as defined in Eq. (34), plotted similarly to the previous two NM measures in Fig. 2.

Figure 4

NM measures for the three representative points P1, P2, and P3 (for details, see the main text), from the parametric plane as a function of time.

Figure 5

NM measures for the same three representative points P1, P2, and P3, as in Fig. 4, as a function of the sample size.

Figure 6

Local divisibility criteria as a function of time for the well-known three points from the parametric plane. The left column shows the quantities, ${\delta }_{1,2,q}$ as defined in Eq. (20); the right column shows its corresponding function $g\left(t\right)$ as defined in Eq. (23). For the intermediate process to be divisible, the quantities on the left must fulfill ${\delta }_2\le {\delta }_q\le {\delta }_1$, and the quantity on the right $g\left(t\right)=0$. We present this figures for points P1 (top panel), P2 (middle panel), and P3 (bottom panel).

Figure 7

Local contractivity criteria as a function of time. In the left figures, we plot the functions which characterize the channel to study how the conditions (32), ${\delta }_q\ge 0$ and ${\delta }_1^{\mathrm{C}}\ge {\delta }_2^{\mathrm{C}}$, are fulfilled or not. On the right we plot ${\sigma }_{\max }$ [Eq. (35)] to study how the measure builds with time, together with its integral (insets). Again, we study points P1, P2, and P3 in the top, middle, and bottom panels.