Contact order governs the onset of entanglement cascades

Abstract

Entangling rates describe a direct entangling channel, but give no information when that channel is forbidden. For finite-dimensional analytic pure-state dynamics, we show that the onset of genuine $(n+1)$-partite entanglement is determined by the contact order $m_\star$ between the physical trajectory and the $S|E$ product manifold. It is the first Taylor order that cannot be reproduced by any product curve, or equivalently the first nonvanishing order of the Fubini--Study distance from the product manifold. We derive a time-ordered recursion that removes curvature-induced kinematic terms and computes $m_\star$ from the Taylor coefficients of $H(t)$. This provides a geometric description of an entanglement cascade, in which new subsystems can join multipartite entanglement only after one or more interaction steps. A symmetry-protected three-qubit model realizes $m_\star=2$, and a three-mode bosonic example shows that genuine tripartite entanglement can arise even when the two newly formed reduced pairs remain separable.