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

2026-05-28

AI Summary

This work investigates finite-size corrections to bipartite quantum mutual information for high-dimensional Haar-random pure states and their geometric origin, transcending the conventional Gaussian approximation framework. Leveraging a central limit theorem for hyperspherical projections, the authors reconstruct the Beta distribution of subsystem occupation probabilities and Lubkin’s purity formula from elementary hyperspherical moments, thereby deriving a complete 1/N asymptotic expansion of the mutual information. They uncover, for the first time, a Bernoulli factorization structure in higher-order corrections—where all odd-order terms vanish—and decompose the leading correction into an su(d_A)⊗su(d_B) quantum coherence contribution and a Cartan⊗Cartan classical probability term, linking these to the generalized Bloch representation and Schur’s majorization theorem. This yields a non-perturbative closed-form expression for the typical mutual information, ⟨I(A:B)⟩ = (d_A²−1)(d_B²−1)·𝒢(d_A,d_B,d_E), which exactly reproduces the Bernoulli series and quantifies the platykurtic tail suppression induced by finite-size effects.

Abstract

Starting from the exact Projected Central Limit Theorem on hyperspheres, we rederive the Beta distribution for subsystem occupation probabilities and Lubkin's purity formula from elementary hyperspherical moments, quantifying the finite-size ``platykurtic'' suppression of tails relative to the Gaussian approximation used in standard eigenstate-thermalization and typicality treatments. Our main new result concerns the bipartite quantum mutual information $\langle I(A{:}B)\rangle$ for Haar-random pure states. We show that its full asymptotic expansion in $1/N$ admits a Bernoulli-factorized form in which every order $k \ge 1$ carries the symmetric factor $(d_A^{2k}-1)(d_B^{2k}-1)$ and all higher odd-order corrections vanish identically. Through an exact algebraic reorganization of Page's formula (conjectured in Ref.~\cite{Page1993} and subsequently proven~\cite{Foong1994, SanchezRuiz1995, Sen1996}), we establish that the leading finite-size correction separates into a dominant $\mathfrak{su}(d_A) \otimes \mathfrak{su}(d_B)$ bipartite quantum coherence contribution $(d_A^2 - 1)(d_B^2 - 1)/(2N)$ and a subtracted classical-probability (Cartan $\otimes$ Cartan) contribution $(d_A - 1)(d_B - 1)/(2N)$, and we trace this separation to the difference between diagonal and eigenvalue entropies via Schur's majorisation theorem, with the dimensional counts $(d-1)$ and $(d^2-1)$ acquiring meaning through the Cartan structure of the generalised Bloch decomposition. These results admit a single non-perturbative closed form: the exact typical mutual information factors as $\langle I(A{:}B)\rangle = (d_A^2-1)(d_B^2-1)\,\mathcal{G}(d_A,d_B,d_E)$, with $\mathcal{G}$ given by an explicit Bose--Einstein integral whose asymptotic expansion in $1/N$ reproduces the Bernoulli series.

Problem

• bipartite entanglement
• quantum mutual information
• typicality
• finite-size correction
• Haar-random states

Innovation

• Projected Central Limit Theorem
• Bipartite Quantum Mutual Information
• Finite-Size Corrections
• Bernoulli Factorization
• Generalized Bloch Decomposition