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

• URL: http://arxiv.org/abs/2605.29725v2
• Date: Thu, 04 Jun 2026 13:46:02 GMT
• Title: Non-Perturbative Closed Form for the Typical Bipartite Mutual Information of Haar-Random States
• Abstract summary: This quantity admits a single non-perturbative closed form.<n>$langle I(A:B)rangle = (d_A2-1)(d_B2-1),mathcalG(d_A,d_B,d_E)$, where $mathcalG$ is given by an explicit convergent integral over a Bose--Einstein kernel.
• Score: 7.7281318046635015
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $ζ(1{-}2k)$; 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.

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