geometric-typicality-entanglement

## 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