Temporal entanglement transition in chaotic quantum many-body dynamics

TL;DR

This work clarifies the relationship between temporal entanglement (TE) of the influence matrix and non-Markovian memory in chaotic quantum baths. By analyzing a structureless random-unitary bath, dual-unitary kicked Ising circuits, and generic Floquet KIMs, it shows that TE exhibits a volume-law regime controlled by the ratio $r=T/b$ (or $1/v_B$ dynamically) and undergoes a sharp transition to area-law scaling under a finite coarse-graining density $n_{ m cg}^$. The key finding is that non-Markovian memory is encoded in highly nonlocal temporal correlations that are largely irrelevant for few-point correlators, allowing low-frequency physics to be captured by a compressed IM. Practically, this implies that an area-law IM, realizable as a matrix-product state with modest bond dimension, suffices to describe local observable dynamics in chaotic baths, with strong implications for efficient simulation and understanding of thermalization in many-body quantum systems.

Abstract

Temporal entanglement (TE) of an influence matrix (IM) has been proposed as a measure of complexity of simulating dynamics of local observables in a many-body system. Foligno et al. [Phys. Rev. X 13, 041008 (2023)] recently argued that the TE in chaotic 1d quantum circuits obeys linear (volume-law) scaling with evolution time. To reconcile this apparent high complexity of IM with the rapid thermalization of local observables, here we study the relation between TE, non-Markovianity, and local temporal correlations for chaotic quantum baths. By exactly solving a random-unitary bath model, and bounding distillable entanglement between future and past degrees of freedom, we argue that TE is extensive for low enough bath growth rate, and it reflects genuine non-Markovianity. This memory, however, is entirely contained in highly complex temporal correlations, and its effect on few-point temporal correlators is negligible. An IM coarse-graining procedure, reducing the allowed frequency of measurements of the probe system, results in a transition from volume- to area-law TE scaling. We demonstrate the generality of this TE transition in 1d circuits by analyzing the kicked Ising model analytically at dual-unitary points, as well as numerically away from them. This finding indicates that dynamics of local observables are fully captured by an area-law IM. We provide evidence that the compact IM MPS obtained via standard compression algorithms accurately describes local evolution.

Figures (15)

• Figure 1: Schematic of a general influence matrix (IM) and temporal entanglement (TE) transition. (a) The IM state is obtained by preparing $T$ qudit pairs (each comprising a "system" qudit $S_\tau$ and a "reference" qudit $Q_\tau$) in maximally entangled Bell states, and then sequentially bringing the qudits $S_1$, $S_2$, …, $S_T$ to interact with a given bath. The IM state is the state of these $2T$ qudits at the end of the procedure (i.e., the state obtained by tracing over the bath degrees of freedom). (b,c) IM before (b) and after (c) the coarse-graining operation, which amounts to inserting unitary evolution of the probe qudit at a fraction of the time steps. As we show, beyond a critical coarse-graining density, TE undergoes a transition from volume-law to area-law (see main text for details).
• Figure 2: (a) IM of the constrained type, arising from probe-bath interactions in product operator form $U=e^{-i H_{\rm probe} \otimes H_{\rm bath}}$. We use here the diagonal tensor notation as in Ref. lerose2021Influence. (b) The IM after coarse-graining procedure with $n_{\rm cg}=1/2$.
• Figure 3: Maximum temporal Rényi-2 TE for the structureless random unitary bath model, plotted as a function of $r$ for different bath sizes $b=\log_2 \mathcal{D}_B$, fixed probe system dimension $d=2$, and pure bath initial state. The dashed line illustrates the analytical prediction, see Eq. (\ref{['eq:Renyi_2_pure']}). The green curve gives a lower bound on distillable entanglement measured in units of $b$. Inset: Rényi-2 TE for $r=1$ and coarse-graining parameters $rn_{\rm cg}=1, 2/3, 1/2$. We observe the transition from volume-law to area-law scaling for $r^\star=1/2$.
• Figure 4: Illustration of the IMs of the static (a) and dynamically growing (b) bath toy models, for $T=6$ time steps and $b=6$ bath qudits (a) or $b_{\text{max}}=6$ maximal number of bath qudits (b), with $v_B=1$. Panel (c): Slope of TE scaling as a function of $v_B$ for several values of $T$, showing the convergence to a limiting slope as $T\to\infty$ and a transition from volume-law to area-law TE scaling at the critical value $v_B^\star=1$.
• Figure 5: Illustration of the estimate of distillable entanglement. (a) We divide the temporal system into past $\mathcal{P}$ and future $\mathcal{F}$ subsystems, and think of evolution as consisting of two steps. (b) (top) In the first step, the bath gets entangled with the $\mathcal{P}$ qubits. (bottom) There exists a distillation protocol, acting only on the $\mathcal{P}$ degrees of freedom (blue box), which distills $N_{\mathcal{B}P}$ Bell pairs, provided the bath is sufficiently large, $b>2p$. (c) In the second step, the Bell-pair partners in $\mathcal{B}$ get entangled with $\mathcal{F}$ qubits. As explained in the text, provided $\mathcal{F}$ is large enough, this allows to distill Bell pairs between $\mathcal{P}$ and $\mathcal{F}$ via a distillation protocol acting on $\mathcal{F}$ only (orange box).

Theorems & Definitions (3)

• Lemma C.1: Decoupling lemma
• Lemma C.2: Lévy concentration inequality
• Definition C.1: Concentration function