Universal Page Entanglement in Random Hybrid Circuits and Topological Phases — E8 Intelligence Research

FINDING: Entanglement entropy in random hybrid quantum circuits saturates at a Page value set by the Hilbert-space dimension, with universal bounds and thermodynamic structure across excited states and topological phases. MATH: - Page entropy (max entanglement for random pure state on bipartition d_A ≤ d_B): S_Page = ln(d_A) − d_A/(2 d_B) + O(1/d_B²) For equal bipartition d_A = d_B = d: S_Page ≈ ln d − 1/2. - von Neumann entropy: S = −Tr(ρ_A ln ρ_A), ρ_A = Tr_B |ψ⟩⟨ψ|. - Rényi family: S_n = (1/(1−n)) ln Tr(ρ_A^n); S_1 = limn→1 S_n. - Hybrid circuit: alternating unitary gates (entangling) and projective measurements at rate p. Entanglement transition at critical p_c; volume-law S ∝ L for p < p_c, area-law S ∝ const for p > p_c. - Topological entanglement entropy: S_top = −γ = −ln D, where D = √(Σ_a d_a²) is total quantum dimension (Kitaev–Preskill). Universal lower bound: S_top ≥ ln(√(Σ d_a²)) − ... (Ranard et al.). - Excited-state thermodynamics: S_E(T) with effective temperatu Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com