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arXiv:2606.30941

Revisiting the Page curve and its moments. A combinatorial approach

Gero von Gersdorff

quant-phhep-thmath-phSubmitted Jun 29, 2026• Announced Jul 1, 2026• 0 citations
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Abstract

We revisit the calculation of the von Neumann (or "entanglement") entropy of a subsystem of a pure quantum state, under the assumption that the latter is drawn at random from a uniform distribution on the full Hilbert space. We derive simple and closed expressions for all power moments, from which the moments of the entropy can be computed by simple differentiation. Our approach (different from the usual one based on random matrix theory and Laguerre polynomials) makes use of Schur-Weyl duality and the character theory of the symmetric group $S_N$ . The paper is self-contained, providing all the necessary mathematical background.

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Page curve for fermionic Gaussian states

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In a seminal paper, Page found the exact formula for the average entanglement entropy for a pure random state. We consider the analogous problem for the ensemble of pure fermionic Gaussian states, which plays a crucial role in the context of random free Hamiltonians. Using recent results from random matrix theory, we show that the average entanglement entropy of pure random fermionic Gaussian states in a subsystem of ${N}_{A}$ out of $N$ degrees of freedom is given by ${\ensuremath{\langle}{S}_{A}\ensuremath{\rangle}}_{\mathrm{G}}=(N\ensuremath{-}\frac{1}{2})\mathrm{\ensuremath{\Psi}}(2N)+(\frac{1}{4}\ensuremath{-}{N}_{A})\mathrm{\ensuremath{\Psi}}(N)+(\frac{1}{2}+{N}_{A}\ensuremath{-}N)\mathrm{\ensuremath{\Psi}}(2N\ensuremath{-}2{N}_{A})\ensuremath{-}\frac{1}{4}\mathrm{\ensuremath{\Psi}}(N\ensuremath{-}{N}_{A})\ensuremath{-}{N}_{A}$, where $\mathrm{\ensuremath{\Psi}}$ is the digamma function. Its asymptotic behavior in the thermodynamic limit is given by ${\ensuremath{\langle}{S}_{A}\ensuremath{\rangle}}_{\mathrm{G}}=N(log2\ensuremath{-}1)f+N(f\ensuremath{-}1)log(1\ensuremath{-}f)+\frac{1}{2}f+\frac{1}{4}log(1\ensuremath{-}f)\phantom{\rule{0.16em}{0ex}}+\phantom{\rule{0.16em}{0ex}}O(1/N)$, where $f={N}_{A}/N\ensuremath{\le}1/2$. Remarkably, its leading order agrees with the average over eigenstates of random quadratic Hamiltonians with number conservation, as found by \L{}yd\ifmmode \dot{z}\else \.{z}\fi{}ba, Rigol, and Vidmar. Finally, we compute the variance in the thermodynamic limit, given by the constant ${lim}_{N\ensuremath{\rightarrow}\ensuremath{\infty}}{(\mathrm{\ensuremath{\Delta}}{S}_{A})}_{\mathrm{G}}^{2}=\frac{1}{2}[f+{f}^{2}+log(1\ensuremath{-}f)]$.

arXiv:2103.05416•Physical review. B./Physical review. B•Jun 29, 2021•63 citations
We revisit the calculation of the von Neumann (or "entanglement") entropy of a subsystem of a pure quantum state, under the assumption that the latter is drawn at random from a uniform distribution on the full Hilbert space. We derive simple and closed expressions for all power moments, from which the moments of the entropy can be computed by simple differentiation. Our approach (different from the usual one based on random matrix theory and Laguerre polynomials) makes use of Schur-Weyl duality and the character theory of the symmetric group SNS_NSN​ . The paper is self-contained, providing all the necessary mathematical background.
The Tsallis entropy is a useful one-parameter generalization to the standard von Neumann entropy in quantum information theory. In this work, we study the variance of the Tsallis entropy of bipartite quantum systems in a random pure state. The main result is an exact variance formula of the Tsallis entropy that involves finite sums of some terminating hypergeometric functions. In the special cases of quadratic entropy and small subsystem dimensions, the main result is further simplified to explicit variance expressions. As a byproduct, we find an independent proof of the recently proven variance formula of the von Neumann entropy based on the derived moment relation to the Tsallis entropy.
Bosonic Gaussian states are a special class of quantum states in an infinite dimensional Hilbert space that are relevant to universal continuous-variable quantum computation as well as to near-term quantum sampling tasks such as Gaussian Boson Sampling. In this work, we study entanglement within a set of squeezed modes that have been evolved by a random linear optical unitary. We first derive formulas that are asymptotically exact in the number of modes for the Rényi-2 Page curve (the average Rényi-2 entropy of a subsystem of a pure bosonic Gaussian state) and the corresponding Page correction (the average information of the subsystem) in certain squeezing regimes. We then prove various results on the typicality of entanglement as measured by the Rényi-2 entropy by studying its variance. Using the aforementioned results for the Rényi-2 entropy, we upper and lower bound the von Neumann entropy Page curve and prove certain regimes of entanglement typicality as measured by the von Neumann entropy. Our main proofs make use of a symmetry property obeyed by the average and the variance of the entropy that dramatically simplifies the averaging over unitaries. In this light, we propose future research directions where this symmetry might also be exploited. We conclude by discussing potential applications of our results and their generalizations to Gaussian Boson Sampling and to illuminating the relationship between entanglement and computational complexity.
Quantum entanglement plays a crucial role in quantum information, quantum teleportation and quantum computation. The information about the entanglement content between subsystems of the composite system is encoded in the Schmidt eigenvalues. We derive here closed expressions for the spectral density of Schmidt eigenvalues for all three invariant classes of random matrix ensembles. We also obtain exact results for average von Neumann entropy. We find that maximum average entanglement is achieved if the system belongs to the symplectic invariant class.
This tutorial develops the Page curve program from quantum entanglement and Hawking radiation to holographic entropy, replica wormholes, and random matrix theory. We derive the normalized random-state benchmark and its Marchenko–Pastur limit, explaining the roles of non-crossing partitions and spectral resolvents. We then distinguish averaging conditional entropies from calculating the entropy of a state with fluctuating energy or area. Gaussian broadening, finite reservoirs, and a quartic critical model illustrate how the preparation and the fluctuation law enter this distinction. Exact finite-dimensional examples separate averaged spectra from ensemble fluctuations and radiation data for one input from information about a quantum channel. Normalized annealed and quenched Rényi entropies have a common von Neumann limit, while their finite-order difference and connected replica moments measure additional ensemble information. Throughout, explicit calculations are distinguished from gravitational applications that require further assumptions. The resulting account provides a self-contained introduction to the established methods and to the remaining questions about microscopic information recovery.
We analyze the properties of entangled random pure states of a quantum system partitioned into two smaller subsystems of dimensions N and M. Framing the problem in terms of random matrices with a fixed-trace constraint, we establish, for arbitrary N≤M, a general relation between the n-point densities and the cross moments of the eigenvalues of the reduced density matrix, i.e., the so-called Schmidt eigenvalues, and the analogous functionals of the eigenvalues of the Wishart-Laguerre ensemble of the random matrix theory. This allows us to derive explicit expressions for two-level densities, and also an exact expression for the variance of von Neumann entropy at finite N,M. Then, we focus on the moments E{K^{a}} of the Schmidt number K, the reciprocal of the purity. This is a random variable supported on [1,N], which quantifies the number of degrees of freedom effectively contributing to the entanglement. We derive a wealth of analytical results for E{K^{a}} for N=2 and 3 and arbitrary M, and also for square N=M systems by spotting for the latter a connection with the probability P(x_{min}^{GUE}≥sqrt[2N]ξ) that the smallest eigenvalue x_{min}^{GUE} of an N×N matrix belonging to the Gaussian unitary ensemble is larger than sqrt[2N]ξ. As a by-product, we present an exact asymptotic expansion for P(x_{min}^{GUE}≥sqrt[2N]ξ) for finite N as ξ→∞. Our results are corroborated by numerical simulations whenever possible, with excellent agreement.
In a seminal paper, Page found the exact formula for the average entanglement entropy for a pure random state. We consider the analogous problem for the ensemble of pure fermionic Gaussian states, which plays a crucial role in the context of random free Hamiltonians. Using recent results from random matrix theory, we show that the average entanglement entropy of pure random fermionic Gaussian states in a subsystem of NA{N}_{A}NA​ out of NNN degrees of freedom is given by ${\ensuremath{\langle}{S}_{A}\ensuremath{\rangle}}_{\mathrm{G}}=(N\ensuremath{-}\frac{1}{2})\mathrm{\ensuremath{\Psi}}(2N)+(\frac{1}{4}\ensuremath{-}{N}_{A})\mathrm{\ensuremath{\Psi}}(N)+(\frac{1}{2}+{N}_{A}\ensuremath{-}N)\mathrm{\ensuremath{\Psi}}(2N\ensuremath{-}2{N}_{A})\ensuremath{-}\frac{1}{4}\mathrm{\ensuremath{\Psi}}(N\ensuremath{-}{N}_{A})\ensuremath{-}{N}_{A}$, where $\mathrm{\ensuremath{\Psi}}$ is the digamma function. Its asymptotic behavior in the thermodynamic limit is given by ${\ensuremath{\langle}{S}_{A}\ensuremath{\rangle}}_{\mathrm{G}}=N(log2\ensuremath{-}1)f+N(f\ensuremath{-}1)log(1\ensuremath{-}f)+\frac{1}{2}f+\frac{1}{4}log(1\ensuremath{-}f)\phantom{\rule{0.16em}{0ex}}+\phantom{\rule{0.16em}{0ex}}O(1/N)$, where $f={N}_{A}/N\ensuremath{\le}1/2$. Remarkably, its leading order agrees with the average over eigenstates of random quadratic Hamiltonians with number conservation, as found by \L{}yd\ifmmode \dot{z}\else \.{z}\fi{}ba, Rigol, and Vidmar. Finally, we compute the variance in the thermodynamic limit, given by the constant ${lim}_{N\ensuremath{\rightarrow}\ensuremath{\infty}}{(\mathrm{\ensuremath{\Delta}}{S}_{A})}_{\mathrm{G}}^{2}=\frac{1}{2}[f+{f}^{2}+log(1\ensuremath{-}f)]$.