Abstract
In a quantum system in a pure state, a subsystem generally has a nonzero entropy because of entanglement with the rest of the system. Is the average entanglement entropy of pure states also the typical entropy of the subsystem? We present a method to compute the exact formula of the momenta of the probability $P(S_A) \mathrm{d}S_A$ that a subsystem has entanglement entropy $S_A$. The method applies to subsystems defined by a subalgebra of observables with a center. In the case of a trivial center, we reobtain the well-known result for the average entropy and the formula for the variance. In the presence of a nontrivial center, the Hilbert space does not have a tensor product structure and the well-known formula does not apply. We present the exact formula for the average entanglement entropy and its variance in the presence of a center. We show that for large systems the variance is small, $\Delta S_A/\langle{S_{A}}\rangle\ll 1$, and therefore the average entanglement entropy is typical. We compare exact and numerical results for the probability distribution and comment on the relation to previous results on concentration of measure bounds. We discuss the application to physical systems where a center arises. In particular, for a system of noninteracting spins in a magnetic field and for a free quantum field, we show how the thermal entropy arises as the typical entanglement entropy of energy eigenstates.
Highlights
In a seminal paper [1], Page showed that, when an isolated quantum system is in a random pure state, the average entropy of a subsystem is close to maximal
In this paper we address the question: Is the average entanglement entropy of pure states hSAi the typical entropy of the subsystem? To illustrate the significance of this question, let us consider for instance the gas in a room held at fixed temperature
Building on [1,38,39,40,41,42,43], we develop methods to compute the exact formula for the average entanglement entropy and the moments mn 1⁄4 hðSA − hSAiÞni of the probability distribution PðSAÞdSA of finding a pure state with entanglement entropy SA
Summary
In a seminal paper [1], Page showed that, when an isolated quantum system is in a random pure state, the average entropy of a subsystem is close to maximal. After presenting a detailed derivation of the Page curve and its variance, we discuss the application of our results to a model system where a nontrivial center arises: we determine the Page curve of energy eigenstates of a paramagnetic solid in a magnetic field (Fig. 1), and show how thermal properties of a subsystem arise from entanglement with the rest of the system. A simple example of subsystems with a center is provided by the energy eigenspace in a noninteracting system with Hamiltonian H 1⁄4 HA þ HB In this case the eigenspace HðEÞ ⊂ H has the structure of a direct sum of tensor products.
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