Universality in volume law entanglement of pure quantum states

TL;DR

The paper tackles how entanglement entropy in pure quantum states deviates from the simple volume law as subsystem size grows, linking this behavior to thermodynamics and experimental observations. It derives a universal two-parameter formula for the second Rényi Page curve in canonical thermal pure quantum states, showing that the curve is governed by a slope a(β) and an offset K(β) and that the universal form extends to scrambled pure states beyond cTPQ. The authors demonstrate the formula's universality by fitting it to non-integrable energy eigenstates and post-quench states, while integrable eigenstates deviate, providing a diagnostic for chaos. They further apply the framework to ETH–MBL transitions, performing finite-size scaling that yields improved estimates of the critical exponent and critical disorder strength, highlighting the practical value of the Page-curve fitting in analyzing thermalization and localization in many-body systems.

Abstract

A pure quantum state can fully describe thermal equilibrium as long as one focuses on local observables. Thermodynamic entropy can also be recovered as the entanglement entropy of small subsystems. When the size of the subsystem increases, however, quantum correlations break the correspondence and cause a correction to this simple volume-law. To elucidate the size dependence of the entanglement entropy is of essential importance in linking quantum physics with thermodynamics, and in addressing recent experiments in ultra-cold atoms. Here we derive an analytic formula of the entanglement entropy for a class of pure states called cTPQ states representing thermal equilibrium. We further find that our formula applies universally to any sufficiently scrambled pure states representing thermal equilibrium, i.e., general energy eigenstates of non-integrable models and states after quantum quenches. Our universal formula can be exploited as a diagnostic of chaotic systems; we can distinguish integrable models from chaotic ones and detect many-body localization with high accuracy.

Figures (5)

• Figure 1: A schematic picture of our setup. The second Rényi Page curve for pure states, $S_2(\ell)$, follows the volume law when $\ell$ is small, but gradually deviates from it as $\ell$ grows. At the middle, $\ell=L/2$, the maximal value is obtained, where the deviation from the volume law is $\ln 2$ (see the Result section). Past the middle $\ell=L/2$, it decreases toward $\ell=L$ and becomes symmetric under $\ell \leftrightarrow L - \ell$.
• Figure 2: Second Rényi Page curve in cTPQ states. The dots represent the second Rényi Page curves in the cTPQ states of the spin system \ref{['TPQ_model']} at an inverse temperature $\beta =4$ calculated by Eq. \ref{['TPQ_S2']} for various system sizes $L$. The lines are the fits by Eq. \ref{['FitFunc']} for the numerical data. The inset shows the fitted values of $\ln a$, $S_2(L/2)/(L/2)$, and the average slope of the curve between $\ell=1$ and $\ell=5$. The dotted lines are the extrapolations to $L \to \infty$ by $1/L$ scaling for $\ln a$ and $S_2(L/2)/(L/2)$ and by $1/L^2$ scaling for the average slope.
• Figure 3: Second Rényi Page curve for general energy eigenstates.a, 2RPCs of several energy eigenstates of the non-integrable Hamiltonian, Eq. \ref{['eigenstate_model']} with $\Delta=2$ and $J_2=4$ (dots), and the fits by our formula \ref{['FitFunc']} (lines). The inset shows the energy spectrum of the Hamiltonian, and the arrows indicate the eigenstates presented in the figure. b, Same as figure a for the integrable Hamiltonian ($\Delta=2, J_2=0$). c, Residuals of fits per site $r_i \equiv L^{-1} \sum_{\ell=0}^L (S_2(\ell)_{i, \mathrm{data}} - S_2(\ell)_{i, \mathrm{fit}} )^2$, where $S_2(\ell)_{i, \mathrm{data}}$ is the 2REE of the $i$-th eigenstate and $S_2(\ell)_{i, \mathrm{fit}}$ is a fitted value of it, for all eigenstates of the non-integrable Hamiltonian \ref{['eigenstate_model']} with $\Delta=2$ and $J_2=4$ (we consider only the sector of a vanishing total momentum and magnetization). The eigenstates are sorted in descending order in terms of the residuals, and the horizontal axis represents their percentiles. The fits become better as the size of the system increases. d, Same as figure c for the integrable Hamiltonian ($\Delta=2, J_2=0$). The fits become worse as the size of the system increases.
• Figure 4: Dynamics of the second Rényi Page curve after quantum quench.a, Time evolution of the 2RPC in a non-integrable system (Eq. \ref{['eigenstate_model']} with $\Delta=1$ and $J_2=0.5$) after a quantum quench from the Néel state. The dotted line is the fitting by Eq. (\ref{['FitFunc']}) for the time average of $S_2(\ell)$. The inset shows the dynamics of the 2REE at the center of the system, $S_2(L/2)$. b, Same as figure a for the integrable Hamiltonian ($\Delta=1, J_2=0$). Eq. (\ref{['FitFunc']}) fits the time average well in both a and b.
• Figure 5: Finite-size scaling across the ETH-MBL phase transition.a, Finite-size scaling of $\ln(a)$, extracted from the fitting of the 2RPC of the eigenstates of the Hamiltonian \ref{['mbl_model']}, versus $L^{1/\nu} (h-h_c)$. The estimation of the critical exponent $\nu$ is significantly improved. b, Same finite-size scaling as a for $s_{2, \mathrm{center}} = S_2(L/2)/(L/2)$, which is a conventional estimate of the 2REE per site MBLTransition2015.