Information scrambling and chaos in open quantum systems

Out-of-time-ordered correlators (OTOCs) have been extensively used over the past few years to study information scrambling and quantum chaos in many-body systems. In this paper, we extend the formalism of the averaged bipartite OTOC of Styliaris et al. [Phys. Rev. Lett. 126, 030601 (2021)] to the case of open quantum systems. The dynamics is no longer unitary but it is described by more general quantum channels (trace preserving, completely positive maps). This “open bipartite OTOC” can be treated in an exact analytical fashion and is shown to amount to a distance between two quantum channels. Moreover, our analytical form unveils competing entropic contributions from information scrambling and environmental decoherence such that the latter can obfuscate the former. To elucidate this subtle interplay, we analytically study special classes of quantum channels, namely, dephasing channels, entanglement-breaking channels, and others. Finally, as a physical application we numerically study dissipative many-body spin chains and show how the competing entropic effects can be used to differentiate between integrable and chaotic regimes.

Figure 1

Nonunitary OTOC $G\left(e^{\mathcal{L}t}\right)$ with $\mathcal{L}=i\mathrm{ad}S+\lambda ({\mathcal{D}}_{\mathbb{B}}-I)$, where $S$ is the swap operator and $\mathbb{B}$ is the Bell basis ($d_A=d_B=2$). The different curves correspond to different choices of the dephasing parameter $\lambda .$ Over the timescale ${\lambda }^{-1}$ on which dephasing becomes relevant, the “scrambling entropy” [first term in Eq. (15)] is balanced, and eventually overwhelmed, by the decoherence-induced entropy production [second term in Eq. (15)]. For any fixed time $t$, the OTOC suppression is exponential in the dephasing strength $\lambda .$ Moreover, in sharp contrast with the unitary case, for any $\lambda \ne 0,$ the infinite time limit of the OTOC is vanishing.

Figure 2

Temporal variation of the open OTOC $G\left({\mathcal{E}}_t\right)$ for the TFIM Eq. (30) with $L=6$ spins. The three curves correspond to varying choices of the dissipation strength $\alpha$ in the Lindblad operators, Eq. (28). The chaotic ($g=-1.05,h=0.5$) and integrable ($g=1,h=0$) phases are clearly distinguishable for the $\alpha =0$ case (closed system); however, increasing the dissipation strength to $\alpha =0.05$ makes them fairly indiscernible and destroys the revivals (or fluctuations) characteristic of integrable systems.

Figure 3

Temporal variation of the open OTOC $G\left({\mathcal{E}}_t\right)$ for the XXZ-NNN model Eq. (31) with $L=6$ spins. The three curves correspond to varying choices of the dissipation strength $\alpha$ in the Lindblad operators Eq. (28). The nonintegrable ($J=1,\mathrm{\Delta }=0.5,J=1,{\mathrm{\Delta }}^{\prime }=0.5$) and integrable ($J=1,\mathrm{\Delta }=0=J={\mathrm{\Delta }}^{\prime }$) phases are clearly distinguishable for the $\alpha =0=\gamma$ case (closed system). The integrable model here can be mapped onto free fermions and hence unlike the TFIM case, even after increasing the dissipation strength ($\alpha =0.1=\gamma$), the system demonstrates revivals (or fluctuations) characteristic of integrable systems.

Figure 4

Temporal variation of the individual terms of the open OTOC $G^{\left(1\right)}\left({\mathcal{E}}_t\right)=\frac{d_B}{d^2}Tr\left[S{\mathcal{E}}^{\otimes 2}\left(S_{A{A}^{\prime }}\right)\right]$ and $G^{\left(2\right)}\left({\mathcal{E}}_t\right)=\frac{1}{d^2}Tr\left[S_{A{A}^{\prime }}{\mathcal{E}}^{\otimes 2}\left(S_{A{A}^{\prime }}\right)\right]$ with $G\left({\mathcal{E}}_t\right)=G^{\left(1\right)}\left({\mathcal{E}}_t\right)-G^{\left(2\right)}\left({\mathcal{E}}_t\right)$. The two figures correspond to the integrable and chaotic limits as considered above for the (a) TFIM and (b) XXZ-NNN model with $L=6$ spins, respectively. The dissipation parameters are $\alpha =0.01,\gamma =0.01$. The first term $G^{\left(1\right)}\left({\mathcal{E}}_t\right)$ originates from environmental decoherence and is similar for both the integrable and the chaotic cases. However, the second term, $G^{\left(2\right)}\left({\mathcal{E}}_t\right)$, is clearly distinct for the two phases and can diagnose quantum chaos even in the presence of dissipation.