Measuring temporal entanglement in experiments as a hallmark for integrability

TL;DR

This work addresses how to diagnose integrability and dynamical classes in many-body quantum systems by introducing generalized temporal Rényi entropies, defined via reduced transition matrices $\tau(t)_{O_j}$. It maps these entropies to an experimentally accessible observable using a double-quench protocol on replicated systems, and validates the approach with tensor-network simulations of the 1D TFIM. The authors provide explicit measurement schemes for $S^{\alpha}(t)_{O_j}$, demonstrate a distinct soft-mode signature in integrable dynamics that is suppressed when integrability is broken, and discuss robustness to finite temperature and multiple experimental platforms. This establishes a practical route to experimentally characterize temporal entanglement and differentiates dynamical classes, with potential connections to bulk geometry in holography and broader implications for quantum simulation of out-of-equilibrium matter.

Abstract

We introduce a novel experimental approach to probe many-body quantum systems by developing a protocol to measure generalized temporal entropies. We demonstrate that the recently proposed generalized temporal entropies [Phys. Rev. Research 6, 033021] are equivalent to observing the out-of-equilibrium dynamics of a replicated system induced by a double quench protocol using local operators as probes. This equivalence, confirmed through state-of-the-art tensor network simulations for one-dimensional systems, validates the feasibility of measuring generalized temporal entropies experimentally. Our results reveal that the dynamics governed by the transverse field Ising model integrable Hamiltonian differ qualitatively from those driven by the same model with an additional parallel field, breaking integrability. They thus suggest that generalized temporal entropies can serve as a tool for identifying different dynamical classes. This work represents the first practical application of generalized temporal entropy characterization in one-dimensional many-body quantum systems and offers a new pathway for experimentally detecting integrability. We conclude by outlining the experimental requirements for implementing this protocol with state of the art quantum simulators.

Figures (5)

• Figure 1: a) From top to bottom we sketch the equilibrium states, the reduced density matrices, and the temporal evolution of a 1D many body system by representing them as surfaces that can be interpreted either as idealized representations of the contraction of 2D networks of elementary tensors encoding such states, or as the field theoretical target space-time of the functional path integrals used to define them. In particular, the standard reduced density matrix of a region A, represented in the middle is obtained by a path integral with an horizontal cut along A. b) From top to bottom we represent, The time dependent expectation value of an operator. Here the cylinder is flattened and times runs vertically in the rectangularly shaped path-integral and space horizontally. In the middle we show the two temporal vectors $\bra{L_{O_j}},\,\ket{R}$, generated by the two boundary-path integrals resulting in cutting the original one along a time surface and representing two Feynman-Vernon influence functionals. The time dependent expectation value is just their overlap while their partial contraction give rise to the reduced transition matrices $\tau(t)_{O_j}$ of \ref{['eq:red_tm']}. They are represented at the bottom as the path integral with a partial temporal cut. From from $\tau(t)_{O_j}$ one can extract the generalized temporal entropies using \ref{['eq:gen_reny_entro']}.
• Figure 2: Left) The generalized temporal purity $T^2$ is represented by a path integral on a two sheeted Riemann surface, where one can only pass from the lower sheet to the upper one along the temporal cut. Right) This path integral can be evaluated experimentally by measuring the expectation value of a local operator acting on a replicated system as described by \ref{['eq:quench1']} and \ref{['eq:quench2']}. The two replicas here are represented by two separated raw of individual constituents (orange spheres) one for each replica. The initial states of the replicas are quenched twice. First using an Hamiltonian that does not include any interactions among the replicas, $H^{(1,1)}+ H^{(2,2)}$ represented by yellow shadows. After a time $T-t$ the Hamiltonian is quenched again, and the replicas interact across the cut as described by the new Hamiltonian $H^{(1,2)}+H^{(2,1)}$ that are represented by orange shadows. The system keeps evolving for a time $t$, until the evolution is interrupted, and the desired operators are measured simultaneously on both replicas.
• Figure 3: Generalized temporal Rényi entanglement entropy $T^2(t) _{\ketbra{0}{0}}= \ln \tr\, (\tau_{\ketbra{0}{0}}( t ))^2$ for the quench dynamics starting in a product state and evolved up to $T/J=4$. Results shown are at $g/J=0.4$ and (a)$h/J=0$ (integrable) and (b)$h/J=0.5$ (nonintegrable). We compute $T^2(t)$ for all possible (temporal) bipartitions $0 \leq t \leq T$ leading to the reduced transition matrices $\tau_{\ketbra{0}{0}}( t )$ and plot it against $t$. We obtain $T^2$ in two complementary numerical approaches. Direct calculation of $T^2(t)$ using $\ket{L}$ and $\bra{R}$ vectors via tensor networks simulation in the thermodynamic limit, \ref{['eq:red_tm']} (reduced transition matrix, blue solid lines), and simulating the replica quench protocol one would perform in an actual experiment, \ref{['eq:quench1']}, using $N=40$ (replica method, dashed yellow lines). As expected the entropy vanishes identically at the beginning $t=0$ and at the end of the evolution $t=T$, but it oscillates for bipartitions in between. We observe near perfect agreement between the two methods that just boil down to computing the same object using two different techniques.
• Figure 4: Temporal purities for systems of made by $N=40$ spins in (a) the ferromagnetic regime of the TFIM of \ref{['eq:ham']} for $g/J=0.4$ and $h/J=0$ which is governed by an integrable dynamics and (b) $g/J=0.4$ and $h/J=0.5$ governed by a non-integrable dynamics. We observe a stark contrast between the integrable and non-integrable case with the response in the latter case being highly suppressed and localised around the temporal cut. Performing the Fourier transform in Eq. \ref{['eq:fourier']} of the data in (a) and (b) we obtain the panels (c) and (d), respectively, where the light features in (c) close to $\omega=0$ and $q=0$ unveil the appearance of soft modes in the integrable case which is absent in (d).
• Figure 5: We show in panel (a) the frequency part of the response $\lambda(\omega)$ defined in Eq. \ref{['eq:lambda']} for several values of the symmetry breaking field $h/J$. We see that in the ferromagnetic phase, for $g/J=0.4$, the soft mode disappears as soon as one moves away from the integrable case $h/J=0$. In panel (b) We show the frequency part of the response (with $\sigma_z$ correlators instead of $\sigma_x$) for $g/J = 1.5$, in the paramagnetic phase, for several values of $h/J$. In this case the soft model does not disappear, but it is exponentially suppressed as $h/J$ increases (inset). Similarly, for quench dynamics initialized with a thermal state, we show in panel (c) the same quantity $\lambda(\omega)$ where one can observe that the results rapidly tend towards their ground state counterpart for already moderate values of $\beta J \simeq 1$. This means that these results should be easy to observe in available experiments based on quantum simulators.