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arxiv: 2605.29725 · v2 · pith:3GOKXDTQnew · submitted 2026-05-28 · 🪐 quant-ph · hep-th· math-ph· math.MP· stat.AP

Non-Perturbative Closed Form for the Typical Bipartite Mutual Information of Haar-Random States

Zhi-Wei Wang , Pei-Wen Li , Samuel L. Braunstein This is my paper

Pith reviewed 2026-06-29 06:41 UTC · model grok-4.3

classification 🪐 quant-ph hep-thmath-phmath.MPstat.AP
keywords quantum mutual informationHaar-random statesPage entropyBose-Einstein integralBorel summationnon-perturbative closed formSchur majorization
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The pith

The average bipartite mutual information of Haar-random pure states equals exactly (d_A²-1)(d_B²-1) times a convergent integral over a Bose-Einstein kernel.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives an exact non-perturbative expression for the typical quantum mutual information between two subsystems in a random pure state of a tripartite system. It starts from Page's entropy formula and decomposes it into a diagonal Dirichlet term plus a Schur-majorisation correction that isolates the quantum part of the correlations. The resulting formula factors out the exact dimension prefactor dim[su(d_A)] dim[su(d_B)] and expresses the remainder as a single integral whose asymptotic series in 1/N is divergent but Borel-summed by the integral itself. This also yields an immediate upper bound on the mutual information and makes the separation of classical and quantum contributions explicit.

Core claim

The average bipartite quantum mutual information equals (d_A²-1)(d_B²-1) G(d_A,d_B,d_E), where G is the explicit convergent integral over the Bose-Einstein kernel; the prefactor is exact, the integral is the Borel sum of the divergent 1/N series whose coefficients involve Bernoulli numbers and zeta(1-2k), and the representation follows from an exact decomposition of Page's entropy into diagonal and eigenvalue-correction pieces that cleanly separate classical from quantum correlations.

What carries the argument

The integral G over a Bose-Einstein kernel that assembles the mutual information after the diagonal-Schur decomposition of Page's entropy.

If this is right

  • The mutual information is bounded above by (d_A²-1)(d_B²-1)/(2N) via scale-inversion symmetry of the kernel.
  • The 1/N asymptotic series of the mutual information is divergent and is resummed exactly by the integral representation.
  • The overall factor (d_A²-1)(d_B²-1) is the precise coefficient, not an asymptotic approximation.
  • Classical and quantum correlations separate explicitly once the mutual information is assembled from the decomposed Page entropies.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same kernel structure may appear in other information measures built from Page entropies, such as conditional mutual information or tripartite information.
  • The exact prefactor suggests that the leading volume-law term in random-state entanglement can be isolated without approximation even at finite N.
  • Because the integral is non-perturbative, it supplies a concrete benchmark for numerical tensor-network or Monte-Carlo studies of typical entanglement in large but finite systems.

Load-bearing premise

Page's entropy formula admits an exact decomposition into a diagonal Dirichlet contribution and a Schur-majorisation eigenvalue correction that assembles into the mutual information by cleanly separating classical from quantum correlations.

What would settle it

Numerical quadrature of the integral G for small fixed d_A, d_B, d_E compared against direct Monte-Carlo sampling of the average mutual information over Haar-random states of the same dimensions.

read the original abstract

The average bipartite quantum mutual information $\langle I(A{:}B)\rangle$ of Haar-random pure states can be expressed exactly through Page's formula in terms of digamma functions. We show that this quantity admits a single non-perturbative closed form: $\langle I(A{:}B)\rangle = (d_A^2-1)(d_B^2-1)\,\mathcal{G}(d_A,d_B,d_E)$, where $\mathcal{G}$ is given by an explicit convergent integral over a Bose--Einstein kernel. The overall factor $(d_A^2-1)(d_B^2-1)=\dim[\mathfrak{su}(d_A)]\cdot\dim[\mathfrak{su}(d_B)]$ is exact, not merely asymptotic. The asymptotic expansion of $\mathcal{G}$ in $1/N$ yields a Bernoulli-factorised series whose coefficients involve $\zeta(1{-}2k)$; this series diverges, and our integral is its exact Borel sum. The integral representation also makes $\langle I\rangle < (d_A^2{-}1)(d_B^2{-}1)/(2N)$ manifest via a scale-inversion symmetry of the kernel. Our derivation traces the mutual information's structure to an exact decomposition of Page's entropy into a diagonal (Dirichlet) contribution and a Schur-majorisation eigenvalue correction, whose assembly into the mutual information cleanly separates classical from quantum correlations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript claims that the average bipartite mutual information ⟨I(A:B)⟩ of Haar-random pure states admits the exact closed form ⟨I(A:B)⟩ = (d_A²-1)(d_B²-1) G(d_A,d_B,d_E), where G is an explicit convergent integral over a Bose-Einstein kernel. This is obtained from Page's digamma formula via an exact decomposition of the entropy into a diagonal (Dirichlet) contribution and a Schur-majorisation eigenvalue correction that separates classical from quantum correlations; the prefactor is exact, the integral is the Borel sum of the divergent 1/N series, and a kernel symmetry makes the bound ⟨I⟩ < (d_A²-1)(d_B²-1)/(2N) manifest.

Significance. If the derivation holds, the result supplies a non-perturbative exact expression and an explicit integral representation that renders certain bounds immediate and demonstrates Borel summability of the asymptotic series. The explicit integral form and the parameter-free prefactor are strengths that could facilitate further analytic work on typical entanglement in random states.

major comments (1)
  1. [Abstract] Abstract: the central claim rests on an exact decomposition of Page's entropy into a diagonal (Dirichlet) contribution plus a Schur-majorisation eigenvalue correction whose assembly into the mutual information yields the stated prefactor and integral. The manuscript must supply the explicit steps connecting the digamma expression to this decomposition and to the integral representation of G; without them the exactness of the prefactor and the non-perturbative character of the result cannot be verified.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need to make the derivation steps fully explicit. We agree that this is necessary to allow verification of the exact prefactor and the non-perturbative character of the result. We address the single major comment below and will incorporate the requested material in the revised version.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim rests on an exact decomposition of Page's entropy into a diagonal (Dirichlet) contribution plus a Schur-majorisation eigenvalue correction whose assembly into the mutual information yields the stated prefactor and integral. The manuscript must supply the explicit steps connecting the digamma expression to this decomposition and to the integral representation of G; without them the exactness of the prefactor and the non-perturbative character of the result cannot be verified.

    Authors: We agree that the explicit intermediate steps are required for independent verification. In the revised manuscript we will insert a dedicated subsection (immediately following the recall of Page's digamma formula) that (i) decomposes the average von Neumann entropy into its Dirichlet (diagonal) term and the Schur-majorisation correction arising from eigenvalue ordering, (ii) assembles these contributions into the mutual information I(A:B), (iii) isolates the exact prefactor (d_A²-1)(d_B²-1), and (iv) performs the change of variables that converts the resulting expression into the stated convergent integral over the Bose-Einstein kernel. These steps will be written out in full, with all intermediate identities displayed, so that the exactness of the prefactor and the Borel-summability property become directly verifiable from the digamma starting point. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation re-expresses external Page formula via independent integral representation

full rationale

The paper starts from the established Page entropy formula (external, 1993) and claims an exact decomposition into Dirichlet diagonal plus Schur-majorisation correction that yields the stated integral form for mutual information. No quoted step reduces the final expression to a fitted parameter, self-definition, or load-bearing self-citation chain; the prefactor and Borel-sum property are presented as algebraic consequences of the decomposition applied to the known digamma expression. The derivation is therefore self-contained against the external benchmark and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of Page's formula (a standard result) and the proposed entropy decomposition; no free parameters, new entities, or ad-hoc axioms beyond domain assumptions are indicated in the abstract.

axioms (1)
  • domain assumption Page's formula for the average entropy of subsystems in Haar-random states expressed via digamma functions
    The derivation begins from this established exact result.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Exact Geometric Typicality and Bipartite Entanglement from the Projected Central Limit Theorem on Hyperspheres

    quant-ph 2026-05 unverdicted novelty 7.0

    Derives exact typical bipartite mutual information as (d_A^2-1)(d_B^2-1) times an explicit Bose-Einstein integral G, with all odd-order 1/N corrections vanishing and leading correction separating into su(d_A)⊗su(d_B) ...

Reference graph

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