Validity of the Landauer principle and quantum memory effects via collisional models

We study the validity of the Landauer principle in the non-Markovian regime by means of collisional models where the intracollisions inside the reservoir cause memory effects generating system-environment correlations. We adopt the system-environment correlations created during the dynamical process to assess the effect of non-Markovianity on the Landauer principle. Exploiting an exact equality for the entropy change of the system, we find the condition for the violation of the Landauer principle, which occurs when the established system-environment correlations become larger than a threshold. We then generalize the study to the nonequilibrium situation where the system is surrounded by many reservoirs at different temperatures. Our results, verified through collisional models with Heisenberg-type interactions, suggest that the complexity of the environment does not play a significant role in the qualitative mechanisms underlying the violation of the Landauer principle under non-Markovian processes.

Figure 1

(a) The system $S$ collides with the reservoir qubit $R_1$ which then become correlated to each other. (b) The intracollision between $R_1$ and $R_2$ takes place leading to the correlations among $S,R_1$, and $R_2$, as denoted by the dashed lines in the figure. (c) The ancilla qubit $R_1$ is traced out, while $S$ and $R_2$ are still correlated. Then $S$ collides with $R_2$, which is followed by the intracollision of $R_2-R_3$ in panel (d). The process is then iterated accordingly.

Figure 2

The LHS (solid symbols) and RHS (hollow symbols) of Eq. (4) against the collision number $n$ for different interenvironment collision strength $\mathrm{\Omega }$. The initial temperatures of the system and the reservoir are chosen as $T_S=3\omega$ and $T_R=\omega$, respectively. The coupling constant between the system and the reservoir qubits is set as $J=0.1\omega$.

Figure 3

The heat $\mathrm{\Delta }{\tilde{Q}}_n$ absorbed by the reservoir and its two contributions $\mathrm{\Delta }{\tilde{Q}}_n^{\mathrm{dia}}$ and $\mathrm{\Delta }{\tilde{Q}}_n^{\mathrm{coh}}$ against the number of collisions $n$ for $\mathrm{\Omega }=1.3\omega$. Other parameters are the same as those of Fig. 2.

Figure 4

The established system-environment correlation in terms of $I\left({\rho }_{S{R}_n}\right)$, the entropy production $D\left({\rho }_{S{R}_n}^{\prime }\parallel {\rho }_{S_n}^{\prime }{\rho }_{R_n}\right)$, and the difference of ${\beta }_{R_n}\mathrm{\Delta }{\tilde{Q}}_n-\mathrm{\Delta }{\tilde{S}}_n$ against the collision number $n$ for different interenvironment collision strength $\mathrm{\Omega }$. The other parameters are the same as that given in Fig. 2. When $I\left({\rho }_{S{R}_n}\right)>D\left({\rho }_{S{R}_n}^{\prime }\parallel {\rho }_{S_n}^{\prime }{\rho }_{R_n}\right)$, we have ${\beta }_{R_n}\mathrm{\Delta }{\tilde{Q}}_n<\mathrm{\Delta }{\tilde{S}}_n$, i.e., the Landauer principle is violated.

Figure 5

(a) System $S$ collides in a sequence with qubits $R_1^{\left(1\right)}, R_1^{\left(2\right)}$ in reservoirs ${\mathcal{R}}^{\left(1\right)}, {\mathcal{R}}^{\left(2\right)}$, respectively, so that they get correlated. The established correlations are denoted by dashed lines. (b) The intracollisions of $R_1^{\left(1\right)}-R_2^{\left(1\right)}$ and $R_1^{\left(2\right)}-R_2^{\left(2\right)}$ take place, generating correlations among $S, R_1^{\left(1\right)}, R_1^{\left(2\right)}, R_2^{\left(1\right)}$, and $R_2^{\left(2\right)}$. (c) Reservoir qubits $R_1^{\left(1\right)}$ and $R_1^{\left(2\right)}$ are traced out, while the correlations of $S-R_2^{\left(1\right)}-R_2^{\left(2\right)}$ remain. Then, $S$ collides with both $R_2^{\left(1\right)}$ and $R_2^{\left(2\right)}$ which is followed by intracollisions of $R_2^{\left(1\right)}-R_3^{\left(1\right)}, R_2^{\left(2\right)}-R_3^{\left(2\right)}$ in panel (d).

Figure 6

The LHS (solid symbols) and RHS (hollow symbols) of Eq. (15) against the collision number $n$. The initial temperatures of the system and the two reservoirs are chosen as $T_S=2\omega , T_{R^{\left(1\right)}}=3\omega$, and $T_{R^{\left(2\right)}}=\omega$, respectively. Here we assume the same strengths $\mathrm{\Omega }$ for the intracollisions of $R_n^{\left(1\right)}-R_{n+1}^{\left(1\right)}$ and $R_n^{\left(2\right)}-R_{n+1}^{\left(2\right)}$ and the same coupling constants $J=0.1\omega$ for $S-R_n^{\left(1\right)}$ and $S-R_n^{\left(2\right)}$.

Figure 7

Established system-environment correlations $I\left({\rho }_{S{R}_n^{\left(1\right)}{R}_n^{\left(2\right)}}\right)$, entropy production $D\left({\rho }_{S{R}_n^{\left(1\right)}{R}_n^{\left(2\right)}}^{\prime }\parallel {\rho }_{S_n}^{\prime }{\rho }_{R_n^{\left(1\right)}}{\rho }_{R_n^{\left(2\right)}}\right)$, and the difference ${\sum }_{m=1}^2{\beta }_{R_n^{\left(m\right)}}\mathrm{\Delta }{\tilde{Q}}_n^{\left(m\right)}-\mathrm{\Delta }{\tilde{S}}_n$ as a function of the number of collisions $n$ for different inter-environment collision strengths $\mathrm{\Omega }$. The other parameters are the same as those given in Fig. 6. When $I\left({\rho }_{S{R}_n^{\left(1\right)}{R}_n^{\left(2\right)}}\right)>D\left({\rho }_{S{R}_n^{\left(1\right)}{R}_n^{\left(2\right)}}^{\prime }\parallel {\rho }_{S_n}^{\prime }{\rho }_{R_n^{\left(1\right)}}{\rho }_{R_n^{\left(2\right)}}\right)$, one has ${\sum }_{m=1}^2{\beta }_{R_n^{\left(m\right)}}\mathrm{\Delta }{\tilde{Q}}_n^{\left(m\right)}<\mathrm{\Delta }{\tilde{S}}_n$, i.e., the Landauer principle violation.