Typical entanglement entropy in the presence of a center: Page curve and its variance

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 (SA)dSA that a subsystem has entanglement entropy SA. 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, ∆SA/ SA 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.Introduction.-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. This result plays a central role in the analysis of the black hole information puzzle [2][3][4][5][6][7][8][9], in the quantum foundations of statistical mechanics [10][11][12][13][14][15][16][17][18][19][20][21][22][23], in quantum information theory [24][25][26][27][28][29][30][31] and in the study of the quantum nature of spacetime geometry [32][33][34][35][36][37]. The entanglement entropy, expressed as a function of the subsystem size, is often called the Page curve ( Fig. 1). In a quantum system in a pure state, a subsystem A generally has a nonzero entropy because of entanglement with the rest of the system. In this paper we address the question: Is the average entanglement entropy of pure states S A also 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. The canonical ensemble allows us to compute the average energy of the gas. However, the configuration of molecules in one room is one realization of this ensemble-we are not averaging over rooms. How close to the average is the energy of this realization? In other words, is the average energy typical? In statistical mechanics, we answer this question by computing the variance (∆E) 2 . In the canonical ensemble, we find ∆E/ E 1 and we conclude that the average energy is typical. Here we investigate the typicality of the entanglement entropy S A of a subsystem by studying its average and variance.A well-studied special case of ...

Skewness of von Neumann entanglement entropy

“…This variance formula was firstly proved in [6], and was independently proved in [2] recently, see also [8] for a discussion on the latter proof. In the present work, we focus on the skewness of von Neumann entropy defined as the third standardized moment…”

“…Proving the above formula of the exact third cumulant of von Neumann entropy is the main contribution of this work. Note that the third moment formula (24) has been recently reported in [2], where the authors described the computations that led to the result (24) as "The computation is an herculean task but with the help of Wolfram's Mathematica we are able to simplify the exact formula for µ 3 ". The rest of the paper is to decode the above description by providing a proof to the claimed result (24).…”

“…In practice, a finite number of cumulants can be utilized to construct approximations to the distribution of the entropy, where the higher cumulants describe the tail distribution. The higher cumulants also provide information such as whether the average entropy is typical [2]. In the literature, the mean and variance of von Neumann entropy have been investigated in [1][2][3][4][5][6] among others.…”

“…The higher cumulants also provide information such as whether the average entropy is typical [2]. In the literature, the mean and variance of von Neumann entropy have been investigated in [1][2][3][4][5][6] among others. The focus of this paper is to study the skewness (involves the second and third cumulants) of von Neumann entropy that measures the degree of asymmetry of the distribution.…”

“…We assume without loss of generality that m ≤ n. The reduced density matrix ρ A of the smaller subsystem A is computed by partial tracing of the full density matrix (2) over the other subsystem B (interpreted as the environment) as…”