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arXiv:2505.03696 · Quantum Science and Technology

Average entanglement entropy of a small subsystem in a constrained pure Gaussian state ensemble *

Erik M. Aurell, Lucas Hackl, Mario Kieburg

Published Oct 1, 2025• 0 citations
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Abstract

Abstract We consider ensembles of pure Gaussian states parametrised by single-mode marginals and (optionally) specific mode-mode correlations. Such ensembles provide a model for the final states when isolated quantum systems thermalise, as they can reproduce thermal properties locally, while being globally pure. By an analysis using real replicas and the coherent state representation of Gaussian states we show that the average entanglement entropy of a small subsystem is the same as the von Neumann entropy of a mixed Gaussian state with the same marginals, but no correlations. Finally, we discuss how these ensembles provide a model for Hawking radiation assuming unitary evolution, and discuss some of their properties in relations to the Page curve of Hawking radiation.

DOI
10.1088/2058-9565/ae0e4e

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Abstract We consider ensembles of pure Gaussian states parametrised by single-mode marginals and (optionally) specific mode-mode correlations. Such ensembles provide a model for the final states when isolated quantum systems thermalise, as they can reproduce thermal properties locally, while being globally pure. By an analysis using real replicas and the coherent state representation of Gaussian states we show that the average entanglement entropy of a small subsystem is the same as the von Neumann entropy of a mixed Gaussian state with the same marginals, but no correlations. Finally, we discuss how these ensembles provide a model for Hawking radiation assuming unitary evolution, and discuss some of their properties in relations to the Page curve of Hawking radiation.
A black hole evaporates by Hawking radiation. Each mode of that radiation is thermal. If the total state is nevertheless to be pure, modes must be entangled. Estimating the minimum size of this entanglement has been an important outstanding issue. We develop a new theory of constrained random symplectic transformations, based on the assumptions that the total state is pure and Gaussian with given marginals. In the random constrained symplectic model we then compute the distribution of mode-mode correlations, from which we bound mode-mode entanglement. Modes of frequency much larger than [k_{B}T_{H}(t)/ℏ] are not populated at time t and drop out of the analysis. Among other relatively thinly populated modes (early-time high-frequency modes and/or late modes of any frequency), we find correlations and hence entanglement to be strongly suppressed. Relatively highly populated modes (early-time low-frequency modes) can, on the other hand, be strongly correlated, but a detailed analysis reveals that they are nevertheless very unlikely to be entangled. Our analysis hence establishes that restoring unitarity after a complete evaporation of a black hole does not require any significant quantum entanglement between any pair of Hawking modes. Our analysis further gives exact general expressions for the distribution of mode-mode correlations in random, pure, Gaussian states with given marginals, which may have applications beyond black hole physics.
Abstract Studying the typical entanglement entropy of a bipartite system when averaging over different ensembles of pure quantum states has been instrumental in different areas of physics, ranging from many-body quantum chaos to black hole evaporation. We extend such analysis to open quantum systems and mixed states, where we compute the typical mutual information in a bipartite system averaged over the ensemble of mixed Gaussian states with a fixed spectrum. Tools from random matrix theory and determinantal point processes allow us to compute arbitrary k-point correlation functions of the singular values of the corresponding complex structure in a subsystem for a given spectrum in the full system. In particular, we evaluate the average von Neumann entropy in a subsystem based on the level density and the average mutual information. Those results are given for finite system size as well as in the thermodynamic limit.
Studying the typical entanglement entropy of a bipartite system when averaging over different ensembles of pure quantum states has been instrumental in different areas of physics, ranging from many-body quantum chaos to black hole evaporation. We extend such analysis to open quantum systems and mixed states, where we compute the typical mutual information in a bipartite system averaged over the ensemble of mixed Gaussian states with a fixed spectrum. Tools from random matrix theory and determinantal point processes allow us to compute arbitrary k-point correlation functions of the singular values of the corresponding complex structure in a subsystem for a given spectrum in the full system. In particular, we evaluate the average von Neumann entropy in a subsystem based on the level density and the average mutual information. Those results are given for finite system size as well as in the thermodynamic limit.
In a quantum system in a pure state, a subsystem generally has a non-zero 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? The exact formula for the average ⟨SA⟩\langle{S_{A}}\rangle⟨SA​⟩ was conjectured by Page in 1995 and later proved. Here we compute the exact formula for the average entropy and variance (ΔSA)2=⟨SA2⟩−⟨SA⟩2(\Delta S_A)^2 = \langle{S_A}^2\rangle - \langle{S_A}\rangle{}^2(ΔSA​)2=⟨SA​2⟩−⟨SA​⟩2 both for random pure states and for energy eigenstates of a non-interacting system. We show that for large systems the variance is small, ΔSA/⟨SA⟩≪1\Delta S_A/\langle{S_{A}}\rangle\ll 1ΔSA​/⟨SA​⟩≪1, and therefore the average entanglement entropy is typical. We compare exact and numerical results for the probability distribution P(SA)dSAP(S_A) \mathrm{d}S_AP(SA​)dSA​, and comment on the relation to previous results on concentration of measure bounds. We discuss the application to physical systems in an energy eigenstate. In particular, for a system of non-interacting spins in a magnetic field, we show how the thermal entropy arises as the typical entanglement entropy of a subsystem.
Abstract. Quantum entanglement of pure states of a bipartite system is defined as the amount of local or marginal (i.e. referring to the subsystems) entropy. For mixed states this identification vanishes, since the global loss of information about the state makes it impossible to distinguish between quantum and classical correlations. Here we show how the joint knowledge of the global and marginal degrees of information of a quantum state, quantified by the purities or in general by information entropies, provides an accurate characterization of its entanglement. In particular, for Gaussian states of continuous variable systems, we classify the entanglement of two–mode states according to their degree of total and partial mixedness, comparing the different roles played by the purity and the generalized p−entropies in quantifying the mixedness and bounding the entanglement. We prove the existence of strict upper and lower bounds on the entanglement and the existence of extremally (maximally and minimally) entangled states at fixed global and marginal degrees of information. This results allow for a powerful, operative method to measure mixedstate entanglement without the full tomographic reconstruction of the state. Finally, we briefly discuss the ongoing extension of our analysis to the quantification of multipartite entanglement in highly symmetric Gaussian states of arbitrary 1 × N-mode partitions. 1.
Quantum entanglement of pure states of a bipartite system is defined as the amount of local or marginal (i.e. referring to the subsystems) entropy. For mixed states this identification vanishes, since the global loss of information about the state makes it impossible to distinguish between quantum and classical correlations. Here we show how the joint knowledge of the global and marginal degrees of information of a quantum state, quantified by the purities or, in general, by information entropies, provides an accurate characterization of its entanglement. In particular, for Gaussian states of continuous variable systems, we classify the entanglement of two-mode states according to their degree of total and partial mixedness, comparing the different roles played by the purity and the generalized p-entropies in quantifying the mixedness and bounding the entanglement. We prove the existence of strict upper and lower bounds on the entanglement and the existence of extremally (maximally and minimally) entangled states at fixed global and marginal degrees of information. This results allow for a powerful, operative method to measure mixed-state entanglement without the full tomographic reconstruction of the state. Finally, we briefly discuss the ongoing extension of our analysis to the quantification of multipartite entanglement in highly symmetric Gaussian states of arbitrary 1 × N-mode partitions.