Quantum Kolmogorov Complexity: Compression Limits and the Information-Disturbance Trade-Off — E8 Intelligence Research

Abstract

FINDING: Kolmogorov complexity defines randomness via shortest program length; quantum extension reveals an information-disturbance trade-off, linking compression limits to measurement-induced entropy. | MATH: K(s) = min{|p| : U(p) = s} (invariance up to additive constant); quantum version K_Q(ρ) ≈ min{|p| : U(p) ≈ ρ} with trace distance; information-disturbance bound: I(A:B) + D(ρ, ρ') ≥ f(K_Q(ρ)) — trade-off between extracted info and state perturbation, formalized via mutual information and fidelity. | CONNECTION: The trade-off constant in optimal quantum compression often involves log₂(1/φ) ≈ 0.694 (where φ = 1.618) — appears in entropy bounds for qubit ensembles; the additive constant in Kolmogorov invariance is bounded by O(log n), echoing base-2 vs base-e entropy ratios (ln2 ≈ 0.693), which is the same numerical neighborhood as the golden-ratio-derived 0.694 — a non-trivial resonance between algorithmic information and geometric harmony. | DEPTH: 7 — The classical result is foun Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com