Application of quantum Darwinism to a structured environment

Quantum Darwinism extends the traditional formalism of decoherence to explain the emergence of classicality in a quantum universe. A classical description emerges when the environment tends to redundantly acquire information about the pointer states of an open system. In light of recent interest, we apply the theoretical tools of the framework to a qubit coupled with many bosonic subenvironments. We examine the degree to which the same classical information is encoded across collections of (i) complete subenvironments and (ii) residual “pseudomode” components of each subenvironment, the conception of which provides a dynamic representation of the reservoir memory. Overall, significant redundancy of information is found as a typical result of the decoherence process. However, by examining its decomposition in terms of classical and quantum correlations, we discover classical information to be nonredundant in both cases i and ii. Moreover, with the full collection of pseudomodes, certain dynamical regimes realize opposite effects, where either the total classical or quantum correlations predominantly decay over time. Finally, when the dynamics are non-Markovian, we find that redundant information is suppressed in line with information backflow to the qubit. By quantifying redundancy, we concretely show it to act as a witness to non-Markovianity in the same way as the trace distance does for nondivisible dynamical maps.

Figure 1

Schematic showing two dynamical representations of the model. (i) The original picture provided by Eqs. (2) and (3), where the qubit system $S$ couples to many subenvironments $E_1,E_2,...$ with strengths ${\mathrm{\Omega }}_1,{\mathrm{\Omega }}_2,...$. (ii) An equivalent picture in terms of pseudomodes, labeled $P_1,P_2,...$, which are each damped by independent Markovian reservoirs $R$ at a rate $\mathrm{\Gamma }$. The environment is sampled by constructing fragments out of the bare subenvironments or the pseudomodes, indicated by $E_f$ and $P_f$, respectively.

Figure 2

Time evolution of the qubit Green's function (blue solid curve) from $|c_e{\left(t\right)|}^2={\left|G\left(t\right)\right|}^2{\left|c_e\left(0\right)\right|}^2$ and excited pseudomode population for $c_e\left(0\right)=1$. Parameters are $\mathrm{\Gamma }=\{{10}^{-2},0.1,0.5\}{\mathrm{\Gamma }}_{+}$ (red long-dashed, violet dashed, black dotted curves) for $\mathrm{\Delta }=0$. (a) Weak-coupling case, ${\mathrm{\Gamma }}_{+}=10{\mathrm{\Omega }}_0$. (b) Strong-coupling (to moderate-coupling) case, ${\mathrm{\Gamma }}_{+}={\mathrm{\Omega }}_0$.

Figure 3

Partial information $\langle I\left(f\right)\rangle /S\left({\rho }_S\right)$ between the qubit and subenvironments shown as a function of fraction size $f$ at times ${\mathrm{\Omega }}_{0}t=\{10,15,25,40,50\}$ (black dotted, violet short-dashed, orange dashed, red long-dashed, and blue solid curves), for parameters (a) ${\mathrm{\Gamma }}_{+}={\mathrm{\Omega }}_0$ and (b) ${\mathrm{\Gamma }}_{+}=10{\mathrm{\Omega }}_0$. In the shown examples, the spreading of correlations over time leads to emergent redundant information, indicated by the presence of the flat plateau. The intercept between the black dashed line (shown for $\delta =0.15$) and each curve highlights the average size of a fraction $f_{\delta }=1/R_{\delta }$ [Eq. (23)] containing up to $(1-\delta )S\left({\rho }_S\right)$ information on the qubit pointer states.

Figure 4

Analytical approximations of the partial information (solid curves) plotted against the corresponding numerical results at times ${\mathrm{\Omega }}_{0}t=\{10,15,40\}$ (black circles, red squares, and blue triangles), for $\mathrm{\Delta }=0$. (a), (b) Average of $I\left({\rho }_{{\mathrm{SE}}_f}\right)$ [Eq. (30)]. (c), (d) Average of $I\left({\rho }_{S{P}_f}\right)$ [Eq. (37)] for $\mathrm{\Gamma }={10}^{-3}{\mathrm{\Gamma }}_{+}$. Note the left-hand column is for parameters ${\mathrm{\Gamma }}_{+}={\mathrm{\Omega }}_0$, and the right-hand column is for ${\mathrm{\Gamma }}_{+}=10{\mathrm{\Omega }}_0$. It can be seen that the analytical results fit the numerics accurately within the plateau region at longer times.

Figure 5

Partial information $\langle I\left(f\right)\rangle /S\left({\rho }_S\right)$ between the qubit and pseudomodes shown as a function of fraction size $f$ at times (a) ${\mathrm{\Omega }}_{0}t=\{5,15,25,40\}$, (b) ${\mathrm{\Omega }}_{0}t=\{0.5,1,5,10\}$, and (c), (d) ${\mathrm{\Omega }}_{0}t=\{10,15,25,40,50\}$ ($t$ increases in the same order as each curve in Fig. 3 from dotted to solid). The left-hand column is for strong (moderate) coupling ${\mathrm{\Gamma }}_{+}={\mathrm{\Omega }}_0$ while the right-hand column is for weak coupling ${\mathrm{\Gamma }}_{+}=10{\mathrm{\Omega }}_0$ (all $\mathrm{\Delta }=0$). Unlike case i, total correlations are erased over time as the state ${\rho }_{S{P}_f}$ evolves through a noisy quantum channel. (a), (b) $\mathrm{\Gamma }=0.4{\mathrm{\Gamma }}_{+}$: partial information decays quickly though for strong coupling redundancy features are qualitatively noticeable. (c), (d) $\mathrm{\Gamma }={10}^{-3}{\mathrm{\Gamma }}_{+}$: a classical plateau is present in the long-time limit since the partial information retains approximate asymmetry about $f=1/2$ (indicated by the arrows), except at the boundary $f=1$.

Figure 6

Time evolution of the purity $P\left(t\right)=\text{tr}\left[{\rho }_{\mathrm{SP}}\right]$ shown for parameters $\mathrm{\Delta }=0$, ${\mathrm{\Gamma }}_{+}={\mathrm{\Omega }}_0$ (blue solid, violet dashed curves), and ${\mathrm{\Gamma }}_{+}=10{\mathrm{\Omega }}_0$ (red long-dashed, black dotted curves).

Figure 7

Total partial information (violet solid curve), partial accessible information (blue long-dashed curve), and partial quantum discord (red dashed curve) between the qubit and subenvironments [see Eqs. (39, 40, 41)], plotted over $S\left({\rho }_S\right)$ at time ${\mathrm{\Omega }}_{0}t=50$, with $\mathrm{\Gamma }={10}^{-3}{\mathrm{\Gamma }}_{+}$ ($\mathrm{\Delta }=0$). Snapshots are shown for parameters (a) ${\mathrm{\Gamma }}_{+}={\mathrm{\Omega }}_0$ and (b) ${\mathrm{\Gamma }}_{+}=10{\mathrm{\Omega }}_0$. The sum of the classical and quantum correlations (violet open points) are also shown, indicating the validity of Eq. (44).

Figure 8

Dynamics of the accessible information (blue solid curve) and quantum discord (red dashed curve) between qubit and pseudomodes ($X=P$), taken from Eqs. (40) and (41) with $\mathrm{\Delta }=0$ (solutions in Appendix pp2). Panels in the lefthand column are plotted for ${\mathrm{\Gamma }}_{+}={\mathrm{\Omega }}_0$, while those in the right-hand column are plotted for ${\mathrm{\Gamma }}_{+}=10{\mathrm{\Omega }}_0$. (a), (b) $\mathrm{\Gamma }=0.4{\mathrm{\Gamma }}_{+}$: classical correlations decay on a fast time scale and quickly approach zero. (c), (d) $\mathrm{\Gamma }={10}^{-3}{\mathrm{\Gamma }}_{+}$: classical correlations decay much slower with a large separation of time scales.

Figure 9

Total partial information (violet solid curve), partial accessible information (blue long-dashed curve), and partial discord (red dashed curve) between the qubit and pseudomodes plotted over $S\left({\rho }_S\right)$ at time ${\mathrm{\Omega }}_{0}t=50$, where $\mathrm{\Gamma }={10}^{-3}{\mathrm{\Gamma }}_{+}$. (a), (b) Same parameters as those in Fig. 7, i.e., (a) ${\mathrm{\Gamma }}_{+}={\mathrm{\Omega }}_0$ and (b) ${\mathrm{\Gamma }}_{+}=10{\mathrm{\Omega }}_0$. Open points show the sum of the averaged classical correlations and discord, as in Eq. (44).

Figure 10

Full accessible information [Eq. (40)] and quantum discord [Eq. (41)] for case ii, plotted as a function of $\mathrm{\Gamma }$ at time ${\mathrm{\Omega }}_{0}t=50$ ($\mathrm{\Delta }=0$). The blue solid and violet dashed curves are for ${\mathrm{\Gamma }}_{+}={\mathrm{\Omega }}_0$, while the red long-dashed and black dotted curves are for ${\mathrm{\Gamma }}_{+}=10{\mathrm{\Omega }}_0$.

Figure 11

Redundancy $R_{\delta }\left(t\right)$, Eq. (23), plotted as a function of time for cases i (red triangles) and ii (blue marked curve) with $\delta =0.15$. Insets: Decay rate $\gamma \left(t\right)$ from Eq. (15) for each set of parameters. (a) ${\mathrm{\Gamma }}_{+}={\mathrm{\Omega }}_0$ and $\mathrm{\Delta }=0.05{\mathrm{\Gamma }}_{+}$: the dashed lines indicate times at which the redundancy peaks. (b) ${\mathrm{\Gamma }}_{+}=10{\mathrm{\Omega }}_0$ and $\mathrm{\Delta }=0$: in the weak-coupling limit, redundancy monotonically increases in time.