A Random Unitary Circuit Model for Black Hole Evaporation

TL;DR

This work studies the dynamics of a quantum system $\\mathcal{S}$ of $N$ qudits coupled to a large environment under a continuous-time, $2$-local random unitary circuit model, inspired by black hole evaporation. It contrasts Haar-random internal dynamics with $U(1)$-conserving dynamics, deriving entanglement growth quantified by the second Rényi entropy $S^{(2)}$ and exploring information retrieval via Hayden–Preskill-type protocols using a four-replica formalism. The key findings are the emergence of two time scales, $t_s \sim \ln N$ and $t_p \sim N$, in the Haar case, and Page-like entanglement dynamics with a charge-dependent Page time in the conserved-case, along with quantitative retrieval behavior of injected information from the environment. The work also develops efficient numerical techniques exploiting permutation symmetry and a six-state effective space to access reasonably large system sizes, providing a tractable toy model for black hole information dynamics and potential connections to holography and tensor networks.

Abstract

Inspired by the Hayden-Preskill protocol for black hole evaporation, we consider the dynamics of a quantum many-body qudit system coupled to an external environment, where the time evolution is driven by the continuous limit of certain $2$-local random unitary circuits. We study both cases where the unitaries are chosen with and without a conserved $U(1)$ charge and focus on two aspects of the dynamics. First, we study analytically and numerically the growth of the entanglement entropy of the system, showing that two different time scales appear: one is intrinsic to the internal dynamics (the scrambling time), while the other depends on the system-environment coupling. In the presence of a $U(1)$ conserved charge, we show that the entanglement follows a Page-like behavior in time: it begins to decrease in the middle stage of the "evaporation", and decreases monotonically afterwards. Second, we study the time needed to retrieve information initially injected in the system from measurements on the environment qudits. Based on explicit numerical computations, we characterize such time both when the retriever has control over the initial configuration or not, showing that different scales appear in the two cases.

Figures (14)

• Figure 1: Pictorial representation of the model introduced in Sec. \ref{['sec:model']}. We consider a system $\mathcal{S}$ and an environment $\mathcal{E}$ consisting of $N$ and $M$ qudits respectively. The evolution is driven by the continuous limit of a random quantum circuit which implements a fast-scrambling dynamics for $\mathcal{S}$ with a tunable coupling between $\mathcal{S}$ and $\mathcal{E}$. At each infinitesimal time step $\Delta t$ a random unitary operator $U_{i,j}$ is applied to randomly chosen qudits in $\mathcal{S}$ with probability $p_1=N\lambda_1\Delta t$, while a swap $W_{l,m}$ between a qudit in the system and one in the environment (randomly chosen) is applied with probability $p_2=\lambda_2\Delta t$.
• Figure 2: Rényi entropy dynamics for different subsystems, obtained by solving Eq. \ref{['eq:purtiy_equation']} with $\lambda_2=0$ (no coupling with the environment), $\lambda_1=1$ and local dimension $d=2$. Subfigure $(a)$, main panel: Rényi-$2$ entropy $S^{(2)}_{\kappa N}(t)$ as a function of time, for $N=640$ and $\kappa=1/4$. Inset: the plot shows the difference between $S^{(2)}_{\kappa N}(t)$ and its maximum possible value $\kappa N$ as a function of time and for increasing system sizes $N$. Denoting by $t^{\ast}_{(\kappa N)}$ the amount of time needed before $\mathcal{S}^{(2)}_{\kappa N}(t)-\kappa N$ becomes larger than a small negative constant $\varepsilon$ (cf. the main text), it is clear from the plot that $t^{\ast}_{(\kappa N)}\sim \ln (N)$. Subfigure $(b)$: Rényi-$2$ entropy $S^{(2)}_{n}(t)$ as a function of the subsystem size $n$, for different times.
• Figure 3: Rényi entropy dynamics for different subsystems, obtained by solving Eq. \ref{['eq:purtiy_equation']}, for local dimension $d=2$ and $N=60$. Subfigure $(a)$: Rényi-$2$ entropy $S^{(2)}_{n}(t)$ as a function of the subsystem size $n$, for different times, with $\lambda_1=1$ and $\lambda_2=10$. Subfigure $(b)$: $S^{(2)}_{n}(t)$ as a function of time for subsystems containing $15$ and $N-15=45$ qudits. The parameters are chosen as $\lambda_1=1$, $\lambda_2=1.5$.
• Figure 4: Subfigure $(a)$: time derivative of the Rényi-$2$ entropy $S^{(2)}_{\kappa N}(t)$ as a function of time, for different values of the system size $N$. The constant dotted line corresponds the value $s_{\lambda}$ defined in Eq. \ref{['eq:s_lambda2']}. Subfigure $(b)$: same data shown in the region close to $t_s$ (cf. the main text). In both figures we chose $\kappa=3/4$, local dimension $d=2$ and $\lambda_1=1$, $\lambda_2=1.5$.
• Figure 5: Rényi entropy $S^{(2)}_{n}(t)$ as a function of time for $\lambda_1=1$ and $\lambda_2=2$. The plots correspond to a random evolution in the presence of a global $U(1)$ conserved charge, while the system is initialized as in Eq. \ref{['eq:initial_state']}. Subfigures $(a)$: $S^{(2)}_{\kappa N}(t)$ for different values of the system size $N$ and $\kappa=0.5, 1$. Subfigure $(b)$: $S^{(2)}_{n}(t)$ for $N=60$ and subsystems containing $15$ and $N-15=45$ qubits. Inset: $\ln S^{(2)}_{n}(t)$ for the same values of $n$.