Abstract
A pure quantum state can fully describe thermal equilibrium as long as one focuses on local observables. The 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 mandate a correction to this simple volume law. The elucidation of the size dependence of the entanglement entropy is thus essentially important in linking quantum physics with thermodynamics. Here we derive an analytic formula of the entanglement entropy for a class of pure states called cTPQ states representing equilibrium. We numerically find that our formula applies universally to any sufficiently scrambled pure state representing thermal equilibrium, i.e., energy eigenstates of non-integrable models and states after quantum quenches. Our formula is exploited as diagnostics for chaotic systems; it can distinguish integrable models from non-integrable models and many-body localization phases from chaotic phases.
Highlights
A pure quantum state can fully describe thermal equilibrium as long as one focuses on local observables
We note that we use the term differently from how it is used in the context of quantum gravity, where it denotes the temporal dependence of the entanglement during the formation of a black hole
We show that the 2RPC S2ð‘Þ is parametrized by only two parameters (Eq (5)), and our formula improves the finite-size scaling of the thermodynamic quantities
Summary
A pure quantum state can fully describe thermal equilibrium as long as one focuses on local observables. The second Rényi EE (2REE), one of the variants of the EE, of a pure quantum state was measured in quantum quench experiments using ultra-cold atoms For such pure quantum states, it is believed that the EE of any small subsystem increases in proportion to the size of the subsystem just like the thermodynamic entropy[9]. The curved structure of the size dependence of the EE, which first increases linearly and decreases, universally appears in various excited pure states, for example, energy eigenstates[10] and states after quantum quenches[11,12]. We call this curved structure a Page curve, after D. We first derive the function of the Page curve for canonical thermal pure quantum
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