geometric-typicality-entanglement

name	geometric-typicality-entanglement
description	Exact geometric typicality and bipartite entanglement methodology using projected central limit theorem on hyperspheres. Derives Beta distribution for subsystem occupation, Lubkin purity formula, and Bernoulli-factorized asymptotic expansion of mutual information for Haar-random states. Applicable to quantum information theory, random matrix theory, eigenstate thermalization.
category	quantum-mathematics

Context

This skill extracts reusable methodology from arXiv:2605.29732 (Wang, Li, Braunstein — "Exact Geometric Typicality and Bipartite Entanglement from the Projected Central Limit Theorem on Hyperspheres"). The paper bridges statistical typicality on high-dimensional hyperspheres with quantum entanglement measures.

Core Methodology

1. Projected Central Limit Theorem on Hyperspheres

• Start from uniform distribution on high-dimensional hypersphere S^{N-1}
• Project onto low-dimensional subspace to derive exact Beta distribution for subsystem occupation probabilities
• Quantifies finite-size "platykurtic" suppression of tails relative to Gaussian approximation
• Key insight: exact moments replace Gaussian approximation in eigenstate thermalization treatments

2. Lubkin Purity Formula Re-derivation

• Derive Lubkin's purity formula from elementary hyperspherical moments
• Avoids group-theoretic machinery (Schur-Weyl duality) — purely geometric approach
• Accessible to statisticians and probabilists without quantum information background

3. Bernoulli-Factorized Asymptotic Expansion

• For bipartite quantum mutual information ⟨I(A:B)⟩ of Haar-random pure states:
• Full asymptotic expansion in 1/N admits Bernoulli-factorized form
• Every order k ≥ 1 carries symmetric factor (d_A^{2k}-1)(d_B^{2k}-1)
• All higher odd-order corrections vanish identically
• Key structural result: symmetry between subsystems A and B at every perturbative order

4. Algebraic Reorganization of Page's Formula

• Exact algebraic reorganization separates leading finite-size correction into:
  • Quantum coherence term: (d_A² - 1)(d_B² - 1)/(2N) — dominant, su(d_A) ⊗ su(d_B) structure
  • Classical probability term: (d_A - 1)(d_B - 1)/(2N) — subtracted, Cartan ⊗ Cartan structure
• Trace separation to difference between diagonal and eigenvalue entropies via Schur's majorisation theorem
• Dimensional counts (d-1) and (d²-1) acquire meaning through Cartan structure of generalized Bloch decomposition

5. Non-perturbative Closed Form

• Exact typical mutual information: ⟨I(A:B)⟩ = (d_A²-1)(d_B²-1) · G(d_A, d_B, d_E)
• G given by explicit Bose-Einstein integral
• Asymptotic expansion reproduces full Bernoulli series

Implementation Steps

1. Define hyperspherical geometry: Start with N-dimensional sphere, define projection operator onto subspace of dimension d_A × d_B
2. Compute hyperspherical moments: Use Beta function identities for exact moments of projected coordinates
3. Derive occupation probabilities: Map moments to subsystem occupation Beta distribution
4. Compute purity: Evaluate purity as second moment of projected distribution → Lubkin formula
5. Mutual information expansion: Expand mutual information in powers of 1/N, identify Bernoulli factors
6. Separate quantum/classical terms: Decompose using Cartan subalgebra vs full Lie algebra dimensions
7. Closed form evaluation: Express result as Bose-Einstein integral G(d_A, d_B, d_E)

Pitfalls

• Gaussian approximation trap: Standard treatments use Gaussian CLT approximation; this methodology shows it misses platykurtic tail suppression. For finite-size systems (small N), the exact Beta distribution is essential.
• Odd-order vanishing: The vanishing of all higher odd-order corrections is a non-trivial symmetry result — do not assume corrections exist at every order.
• Dimension counting: The distinction between (d-1) and (d²-1) is subtle but critical — (d-1) counts Cartan subalgebra dimensions (classical), (d²-1) counts full Lie algebra dimensions (quantum coherence).
• Bernoulli factors: The Bernoulli numbers appear in the asymptotic expansion coefficients — ensure correct sign convention (B₁ = -1/2, B₂ = 1/6, etc.).

Verification

• Reproduce Lubkin purity formula for small dimensions (d_A = d_B = 2) using hyperspherical moments
• Verify Bernoulli-factorized form matches known Page formula asymptotics
• Check that odd-order terms beyond first order vanish in the expansion
• Confirm Bose-Einstein integral G reproduces leading-order Page correction

Activation

geometric typicality, bipartite entanglement, projected central limit theorem, hypersphere, Haar random states, Page formula, Lubkin purity, Bernoulli factorization, quantum mutual information, eigenstate thermalization, random matrix theory, 2605.29732