Information scrambling in chaotic systems with dissipation

Chaotic dynamics in closed local quantum systems scrambles quantum information, which is manifested quantitatively in the decay of the out-of-time-ordered correlators (OTOC) of local operators. How is information scrambling affected when the system is coupled to the environment and suffers from dissipation? In this paper, we address this question by defining a dissipative version of OTOC and numerically study its behavior in a prototypical chaotic quantum chain in the presence of dissipation. We find that dissipation leads to not only the overall decay of the scrambled information due to leaking but also structural changes so that the ‘information light cone’ can only reach a finite distance even when the effect of overall decay is removed. Based on this observation we conjecture a modified version of the Lieb-Robinson bound in dissipative systems.

Figure 1

OTOC $F(t,{\sigma }_i^z,{\sigma }_1^z)$ in the chaotic Ising chain (5) with no dissipation (upper left), amplitude damping (upper right), phase damping (lower left), and phase depolarizing (lower right). The dissipation rate is $\mathrm{\Gamma }=0.1$ in these three dissipative channels.

Figure 2

Corrected OTOC $F(t,{\sigma }_i^z,{\sigma }_1^z)/F(t,I,{\sigma }_1^z)$ in the chaotic Ising chain (5) with no dissipation (upper left), amplitude damping (upper right), phase damping (lower left), and phase depolarizing (lower right). The dissipation rate is $\mathrm{\Gamma }=0.1$ in these three dissipative channels.

Figure 3

Corrected OTOC $F(t,{\sigma }_i^z,{\sigma }_1^z)/F(t,I,{\sigma }_1^z)$ recovers some part of the light cone in the channel of phase depolarizing with dissipation rate $\mathrm{\Gamma }=0.1$ and system size $N=24$.

Figure 4

Color plot of the weights of few-body terms in the chaotic Ising chain (5) with $N=12$ spins and dissipation rate $\mathrm{\Gamma }=0.1$. Dashed (dotted) line denotes the total weight of one-body (nearest-neighbor two-body) terms in the operator ${\mathcal{V}}_b^{†}\left(t\right)·{\sigma }_1^z$, while the solid line is the sum of dotted and dashed lines. Black, blue, and red lines are the results for dissipative channels of amplitude damping, phase damping, and phase depolarizing, respectively.

Figure 5

The log-log plot of $d\left(\mathrm{\Gamma }\right)$ and $\mathrm{\Gamma }$.

Figure 6

The log-log plot of $d\left(\mathrm{\Gamma }\right)$ and $\mathrm{\Gamma }$.

Figure 7

The operator norm of the commutator $\parallel [{\mathcal{V}}_b^{†}\left(t\right)·{\sigma }_1^z,{\sigma }_i^z]\parallel$ in the chaotic Ising chain (5) with no dissipation (upper left), amplitude damping (upper right), phase damping (lower left), and phase depolarizing (lower right). The dissipation rate is $\mathrm{\Gamma }=0.2$ in these three dissipative channels.

Figure 8

The corrected operator norm of the commutator $\parallel [{\mathcal{V}}_b^{†}\left(t\right)·{\sigma }_1^z,{\sigma }_i^z]\parallel /\parallel {\mathcal{V}}_b^{†}\left(t\right)·{\sigma }_1^z\parallel$ in the chaotic Ising chain (5) with no dissipation (upper left), amplitude damping (upper right), phase damping (lower left), and phase depolarizing (lower right). The dissipation rate is $\mathrm{\Gamma }=0.2$ in these three dissipative channels.

Figure 9

The decay of the operator norm $\parallel {\mathcal{V}}_b^{†}\left(t\right)·{\sigma }_1^z\parallel$ in different dissipative channels $\left(N=12,\mathrm{\Gamma }=0.2\right)$.