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

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 = \(d_A²-1\)\(d_B²-1\) 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 \(d_A²-1\)\(d_B²-1\)=dim\[mathfrak{su}\(d_A\)\]cdotdim\[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 Irangle < \(d_A²{-}1\)\(d_B²{-}1\)/(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.