Double-Exponential Quasi-Orthogonality: The Geometry of Decoherence

Abstract

A composite system of N local q-level factors has dimension D=q^N. Although at most D vectors can be exactly orthogonal, a fixed squared-overlap tolerance \eps permits M_\eps(D)\gtrsim\exp(c_\eps D) mutually quasi-orthogonal directions. Consequently M_\eps(q^N)\gtrsim\exp(c_\eps q^N): the dimension grows exponentially in N, but its quasi-orthogonal capacity grows doubly exponentially. L\'evy's lemma explains the accompanying concentration of regular observables, and Johnson--Lindenstrauss scaling gives the complementary finite-set account of exponential capacity. For Haar-random pure states the sharper exact law is \mathbb P(|\langle\phi|\psi\rangle|^2\geq\eps)=(1-\eps)^{D-1}, while typical Fubini--Study angles lie within O(D^{-1/2}) of \pi/2: capacity explodes as angular structure homogenizes. We turn these facts into simultaneous overlap and trace-distance bounds for finite decoherence branch families, including mixed environments and collective weak coherences. The results are conditional on typical relative environmental dynamics. They neither select a pointer basis nor identify coherence suppression with readable or redundant records.