Multipartite dynamics beyond bipartite entanglement equilibration

Abstract

Entanglement equilibration need not mark the end of quantum-information dynamics in quantum thermalization. We show that the complete Schmidt spectra of a fixed tripartition can become stationary while its multipartite correlation structure, distinguished by local-unitary invariants, continues to evolve. This separation occurs exactly within a finite window in self-dual kicked-Ising dynamics and asymptotically in Haar-random circuits. We further demonstrate an independently tunable timescale for multipartite dynamics using quantum-error-correcting codes. When coupled to a locally scrambling bath, ordinary entanglement approaches equilibrium through rapid local scrambling, while weak logical--bath coupling drives a slower reorganization of multipartite correlations. These results reveal a deeper distinction than the ordering of equilibration times: bipartite entanglement and multipartite correlation structure can probe distinct physical channels of quantum-information dynamics, rather than merely different stages of the same relaxation process.

The paper asks whether multipartite correlations can evolve after the complete Schmidt spectra of a fixed pure-state tripartition have equilibrated, and whether this evolution can have an independently tunable timescale. Multipartite structure is identified with local-unitary equivalence classes and probed mainly through a four-replica invariant, the genuine tripartite multi-entropy $G$. In a periodic self-dual kicked-Ising chain with $N=3\ell$, dual-unitary contractions establish exactly maximally mixed one-party marginals for $1+\lceil\ell/2\rceil\leq t\leq\ell+1$. State-vector calculations at $N=18$ nevertheless give $G\simeq(0,0.265,0.600,0.316)$ across the four plateau times. For Haar-random Choi circuits, exact two-copy transfer recurrences and signed four-copy calculations establish a related large-local-dimension separation: at fixed $N=18$ and depths 8 and 10, all marginals converge to maximal mixing in average trace norm, while the annealed multi-entropy changes by $2\log d+O(d^{-2})$. A $[[5,1,3]]$ code supplies protected logical dynamics invisible to individual code shares. Perturbation theory gives coherent multipartite oscillations on a $g^{-3}$ timescale, with entropy deviations $O(g^2)$. Coupling the code to a nine-qubit scrambling bath produces measured multipartite response times scaling as $\lambda^{-2.025}$, while dominant ordinary-entropy growth reaches its operational plateau after five periods. An exact averaged logical channel independently yields $\Gamma_L=2\lambda^2+O(\lambda^4)$. The exact kicked-Ising plateau is finite; the Haar result concerns annealed moments, not typical-sample multi-entropy. The local-bath spectra retain slow corrections, and its four-copy closure is approximate. These constructions do not establish generic thermalization or asymptotic multipartite relaxation gaps.