Monogamy and backflow of mutual information in non-Markovian thermal baths

We investigate the dynamics of information among the parties of tripartite systems. We start by proving two results concerning the monogamy of mutual information. The first one states that mutual information is monogamous for generic tripartite pure states. The second shows that, in general, mutual information is monogamous only if the amount of genuine tripartite correlations is large enough. Then, we analyze the internal dynamics of tripartite systems whose parties do not exchange energy. In particular, we allow for one of the subsystems to play the role of a finite thermal bath. As a result, we find a typical scenario in which local information tends to be converted into delocalized information. Moreover, we show that (i) the information flow is reversible for finite thermal baths at low temperatures, (ii) monogamy of mutual information is respected throughout the dynamics, and (iii) genuine tripartite correlations are typically present. Finally, we analytically calculate a quantity capable of revealing favorable regimes for non-Markovianity in our model.

Figure 1

$|{\theta }_{-}\left(t\right)|$ (first column) and $I_3\left(t\right)$ (second column) as a function of time (in log scale) for $N=10$, $c_{1,2,3}=-0.8$, $\hslash ={\omega }_0=1$, $g_A=1$, $g_B=2$, and $g_0=0.1$. The varying parameters are the temperature and the width of the spectral distribution: (a), (b) $\beta =1$ and $\delta =10N$; (c), (d) $\beta =0.1$ and $\delta =10N$; (e), (f) $\beta =0.1$ and $\delta =N$. All parameters are given in arbitrary units. The vertical line in each panel accounts for the decoherence time (29).

Figure 2

Quantum discord $D_{AB}^{⟵}\left(t\right)$ (solid line) and accessible information $J_{AB}^{⟵}\left(t\right)$ (dashed line) as a function of time (in log scale) for the same parameters of Figs. 1, 1. In (a) $c_{1,2,3}=-0.6$ and in (b) $c_{1,2}=-0.6$ and $c_3=-0.5$. The solid vertical lines account for the decoherence time (29), whereas the dashed vertical line in (b) accounts for the sudden-change time (34). The constant value of the accessible information is given by $J_{AB}^{⟵}\left(\right|c_3\left|\right)$.

Figure 3

Non-Markovianity ${\mathcal{N}}_M$ as a function of the inverse temperature $\beta$ and the number of modes $N$ for $\hslash ={\omega }_0=1$, $g_A=1$, $g_B=2$, $g_0=0.1$, and $\delta =10N$. In this simulation we used $t={10}^6$. All parameters are given in arbitrary units. Non-Markovianity is favored by small reservoirs and low temperatures.

Figure 4

(a) $I_3/4$ (solid line) and $\mathfrak{I}$ (dashed line) as a function of $x$ for the state (A1). Monogamy is violated by mixed states with $x\gtrsim 0.43596$. (b) $I_3-\mathfrak{I}$ (yellow thick line) and ${min}_{(A,B,C)}(I_{A:C}+I_{B:C})$ (red dashed line) as a function of $x$ for the state (A1). This simulation illustrates the validity of Eq. (9).