  Finding suitable characterizations of quantum chaos is a major challenge in
many-body physics, with a central difficulty posed by the linearity of the
Schr\"odinger equation. A possible solution for recovering non-linearity is to
project the dynamics onto some variational manifold. The classical chaos
induced via this procedure may be used as a signature of quantum chaos in the
full Hilbert space. Here, we demonstrate analytically a previously heuristic
connection between the Lyapunov spectrum from projection onto the matrix
product state (MPS) manifold and the growth of entanglement. This growth occurs
by squeezing a localized distribution on the variational manifold. The process
qualitatively resembles the Cardy-Calabrese picture, where local perturbations
to a moving MPS reference are interpreted as bosonic quasi-particles. Taking
careful account of the number of distinct channels for these processes recovers
the connection to the Lyapunov spectrum. Our results rigorously establish the
physical significance of the projected Lyapunov spectrum, suggesting it as an
alternative method of characterizing chaos in quantum many-body systems, one
that is manifestly linked to classical chaos.
