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

2026-05-28

AI Summary

This work presents the first exact non-perturbative closed-form expression for the typical bipartite quantum mutual information of Haar-random pure states. By decomposing the Page entropy into a diagonal contribution and a correction term involving Schur-concave eigenvalue functionals, the authors disentangle classical and quantum correlations and uncover a direct connection between mutual information and the dimension of the su algebra. Their approach synergistically combines the Page formula, digamma functions, Schur theory, Borel summation, and Bose–Einstein integral kernels to achieve exact resummation of a divergent asymptotic series. The resulting expression not only yields a rigorous upper bound ⟨I⟩ < (d_A²−1)(d_B²−1)/(2N) but also provides a convergent integral representation that fully reproduces the mutual information’s asymptotic behavior to all orders.

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.

Problem

• quantum mutual information
• Haar-random states
• non-perturbative
• closed form
• bipartite entanglement

Innovation

• non-perturbative
• closed-form
• quantum mutual information
• Haar-random states
• Borel summation