Non-Markovianity through flow of information between a system and an environment

Exchange of information between a quantum system and its surrounding environment plays a fundamental role in the study of the dynamics of open quantum systems. Here we discuss the role of the information exchange in the non-Markovian behavior of dynamical quantum processes following the decoherence approach, where we consider a quantum system that is initially correlated with its measurement apparatus, which in turn interacts with the environment. We introduce a way of looking at the information exchange between the system and environment using the quantum loss, which is shown to be closely related to the measure of non-Markovianity based on the quantum mutual information. We also extend the results of Fanchini et al. [Phys. Rev. Lett. 112, 210402 (2014)] in several directions, providing a more detailed investigation of the use of the accessible information for quantifying the backflow of information from the environment to the system. Moreover, we reveal a clear conceptual relation between the entanglement- and mutual-information-based measures of non-Markovianity in terms of the quantum loss and accessible information. We compare different ways of studying the information flow in two theoretical examples. We also present experimental results on the investigation of the quantum loss and accessible information for a two-level system undergoing a zero temperature amplitude damping process. We use an optical approach that allows full access to the state of the environment.

Figure 1

We consider an initially pure environment $\mathcal{E}$, and an entangled pure state $\mathcal{SA}$. As the system $\mathcal{S}$ evolves free of any direct interaction, the apparatus is interacting with the environment $\mathcal{E}$.

Figure 2

The entropy diagram of the tripartite system composed of the system $\mathcal{S}$, the apparatus $\mathcal{A}$, and the environment $\mathcal{E}$, before and after the interaction. The amount of information will stay the same inside the area enclosed by thick red curves, i.e., $I=\tilde{I}+\tilde{L}=2S\left({\rho }_{\mathcal{S}}\right)$, where $\tilde{I}$ is the mutual information and $\tilde{L}$ is the quantum loss.

Figure 3

The accessible information $J_{\mathcal{S}\tilde{\mathcal{E}}}^{\leftarrow }$ and the inaccessible information ${\delta }_{\mathcal{S}\tilde{\mathcal{E}}}^{\leftarrow }$ in the entropy diagram of the tripartite system $\mathcal{SAE}$ after the interaction of the apparatus $\mathcal{A}$ with the environment $\mathcal{E}$.

Figure 4

(a) Sketch of the experimental setup. The H's are half-wave plates, the Q's are quarter-wave plates, the BD's are beam displacers, PBS is polarizing beam splitter, and DET is single photon detector. The light beams propagate from left to right. (b) Implementation of the amplitude damping channel for the photon polarization, condition $p=0$. (c) Implementation of the amplitude damping channel for the photon polarization, condition $p\ne 0$.

Figure 5

Theoretical plot of the quantum loss $\tilde{L}$ (blue line) and the quantum mutual information $\tilde{I}$ (dashed red line). The insets display the decay rate $\gamma \left(t\right)/{\gamma }_0$ (orange line) as a function of scaled time ${\gamma }_{0}t$, that is being experimentally controlled by $p\left(t\right)\to {\theta }_p$. Experimental points for $\tilde{L}$ and $\tilde{I}$ are shown by black dots and purple triangles, respectively. As (a) demonstrates the monotonicity of information flow in the Markovian regime with $\lambda /{\gamma }_0=3$, (b) displays its nonmonotonous behavior in the non-Markovian regime with $\lambda /{\gamma }_0=0.1.$

Figure 6

Theoretical plot of the accessible information $J_{\mathcal{S}\tilde{\mathcal{E}}}^{\leftarrow }$ (blue line) and the entanglement of formation $E_{\mathcal{S}\tilde{\mathcal{A}}}$ (dashed red line). The insets display the decay rate $\gamma \left(t\right)/{\gamma }_0$ (orange line) as a function of scaled time ${\gamma }_{0}t$, that is being experimentally controlled by $p\left(t\right)\to {\theta }_p$. Experimental points for $J_{\mathcal{S}\tilde{\mathcal{E}}}^{\leftarrow }$ and $E_{\mathcal{S}\tilde{\mathcal{A}}}$ are shown by black dots and purple triangles, respectively. As (a) demonstrates the monotonicity of information flow in the Markovian regime with $\lambda /{\gamma }_0=3$, (b) displays its nonmonotonous behavior in the non-Markovian regime with $\lambda /{\gamma }_0=0.1.$

Figure 7

(a) Plot of the quantum loss $\tilde{L}$ (solid blue line) and the quantum mutual information $\tilde{I}$ (dashed red line). (b) Plot of the accessible information $J_{\mathcal{S}\tilde{\mathcal{E}}}^{\leftarrow }$ (solid blue line) and the entanglement of formation $E_{\mathcal{S}\tilde{\mathcal{A}}}$ (dashed red line). (c) Plot of the nondivisibility criterion $g\left(t\right)$. The dynamical map becomes nondivisible when $g\left(t\right)>0$. In all three plots, we set $\omega =5$ and the initial state is a maximally entangled one.