Revisiting the Page curve and its moments. A combinatorial approach

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.

• The paper introduces an algebraic-combinatorial approach that avoids explicit Haar measure integration to compute the power moments of subsystem entanglement entropy.
• It leverages Schur-Weyl duality and the Murnaghan-Nakayama rule to derive concise, closed-form expressions for the Page curve and its higher moments.
• The approach reveals key analytical properties and scalability, enhancing computational tractability in quantum statistical mechanics and black hole physics.

Algebraic and Combinatorial Approaches to the Page Curve and Its Moments

Introduction

The Page curve, describing the mean entanglement entropy of a subsystem of a random pure quantum state, is central to the statistical exploration of quantum entanglement, the black hole information paradox, and the foundation of quantum statistical mechanics. The referenced paper "Revisiting the Page curve and its moments. A combinatorial approach" (2606.30941) systematically reformulates the computation of the moments of the von Neumann entropy of subsystems for random pure states using an algebraic and combinatorial framework, eschewing the conventional path of random matrix theory.

Problem Formalism and Background

Consider a bipartite quantum system consisting of subsystems A and B with Hilbert space dimensions $m$ and $n$. The focus lies on the reduced density matrix $\rho_A$ after tracing over subsystem B from a Haar-random pure state on the full system. The main quantity of interest is the von Neumann entropy $S_A = -\mathrm{Tr}(\rho_A \log \rho_A)$, which turns into a statistical random variable if the initial pure state is randomly drawn.

Earlier approaches to the average entanglement entropy and its fluctuations—initiated by Lubkin and Page [Page:1993df, Lubkin:1978nch] and further proven via random matrix theory [Foong:1994eja, Sommers:2004xri]—rely on explicit integration over the Haar measure. While they provide closed form expressions for the mean (the Page curve) and in some cases the variance, higher moments required either intricate multiple integrals or combinatorial enumeration.

Algebraic Reformulation: Invariance and Group Theory

The paper develops an algebraic, integration-free approach, leveraging the invariance of the Haar measure under unitary rotations and the underlying symmetric group structure of the tensor powers involved. The key mathematical tools invoked are:

• Schur-Weyl duality: The decomposition of tensor spaces under commuting actions of the symmetric group $S_N$ and the unitary group $\mathrm{U}(d)$.
• Character theory of $S_N$ and the Murnaghan-Nakayama (MN) rule: For combinatorial computation of irreducible characters and associated multiplicities.
• Content polynomial and hook length formulas: For calculating dimensions and weights associated with Young tableaux assigned to symmetric group representations.

This combinatorial setup translates the task of calculating expected traces and products of powers of $\rho_A$ into sums over characters of $S_N$, where explicit expressions are constructed using the MN rule.

Explicit Results for Power Moments

The main technical result is an explicit, closed formula for the power moments of the reduced density matrix:

$$\overline{ \mathrm{Tr} [\rho_A^{\lambda_1}] \cdots \mathrm{Tr} [\rho_A^{\lambda_k}] } = \frac{Z_\lambda}{Z_{(1^N)}}$$

with $n$0 given by

$n$1

where:

• $n$2 is a partition of $n$3,
• $n$4 are the $n$5 characters (computed via the MN rule),
• $n$6 is the content polynomial, and $n$7 is the hook product for the Young diagram $n$8.

For $n$9, corresponding to the moment relevant for the average entropy and Page curve, the sum collapses to hook partitions and yields a single-sum expression. Higher-order mixed moments $\rho_A$0 (variance) and arbitrary multi-partite power moments admit explicit double or multi-dimensional sum expressions, featuring cross ratios that encode the combinatorial contributions of border strip decompositions in MN theory.

Analytical Structure and Continuation

A notable aspect of the method is that while its combinatorial derivation provides expressions for integer moments only, the resulting formulas admit analytic continuation to positive real moments by interpretation of factorial functions as Gamma functions. The paper provides a careful treatment of the uniqueness and monotonicity of the analytic continuation, grounding it in classical moment problems.

The expressions possess several concrete analytical properties:

• The single-moment function $\rho_A$1 can be expressed via terminated hypergeometric functions at integer arguments, and is a rational function in $\rho_A$2.
• All moments are cut off at $\rho_A$3, making the sum efficiently computable for $\rho_A$4 of moderate size.
• Symmetry under exchange $\rho_A$5 is manifest; the formula naturally obeys $\rho_A$6 for the bipartite system.
• For $\rho_A$7 (pure state on subsystem A), all moments reduce correctly to the pure state case.

Comparison with Prior Work and Computational Advantages

While prior combinatorial approaches to the Page curve and its variance exist [Bianchi:2019stn, Liu:2017lem, Collins:2009hos], this work presents an overview and simplification, reducing the depth of nested summations and group integrals required. The authors verify that their method yields identical results to those obtained in prior work but with markedly more concise formulas, improving both analytical tractability and numerical evaluation.

The group-theoretic approach also clarifies the relation between the so-called "replica trick" for entropy calculations in quantum field theory/gravity (which uses analytic continuation in replica number) and the discrete combinatorics of symmetric group representations appearing in quantum many-body problems.

Implications and Future Directions

Physical Relevance:

The results provide algebraically transparent, efficiently computable expressions for the moments of subsystem entropy for generic high-dimensional entangled quantum states, underpinning foundational statements about typicality and thermalization in quantum statistical physics. The formalism is directly relevant to the study of black hole evaporation and the Page curve in quantum gravity, especially in the context of recent advances employing replica wormholes [Penington:2019npb, Almheiri:2019psf].

Combinatorial Insights:

The direct connection to symmetric group character theory opens avenues for employing more advanced tools from algebraic combinatorics and representation theory in entanglement and random matrix theory. The explicit identification of border strip decompositions controlling the growth of entropy fluctuations may find utility in the study of symmetry-resolved entanglement and quantum chaos.

Algorithmic and Analytical Developments:

Given that all higher moments of entropy may be systematically accessed through these algebraic means and that the formulas admit real-argument extension, further work may target the closed-form evaluation of higher cumulants and the systematic study of the full statistical distribution of the entropy. This framework can also be adapted to random mixed states, non-uniform unitary ensembles, or systems with additional symmetries.

Conclusion

This work delivers a comprehensive, self-contained algebraic-combinatorial framework for calculating all power moments of the density matrix and subsequently all moments of the subsystem entanglement entropy for Haar-random bipartite quantum states. By deploying Schur-Weyl duality and the Murnaghan-Nakayama rule, the method supersedes prior approaches founded in explicit integration and yields succinct, analytic formulas suitable for both theoretical exploration and practical computation. The findings are highly relevant for quantum information theory, statistical mechanics, and the study of quantum gravity, and establish a robust foundation for further combinatorial investigations of quantum entropy statistics.

Reference: "Revisiting the Page curve and its moments. A combinatorial approach" (2606.30941).