Entanglement dynamics and Page curves in random permutation circuits

TL;DR

This paper investigates how random permutation circuits affect entanglement in many-qubit systems, deriving bounds and comparing entanglement growth in different circuit models, revealing classical features' impact on quantum entanglement.

Contribution

It provides the first tight bounds on entanglement generated by permutation circuits and compares entanglement Page curves between different circuit ensembles in the thermodynamic limit.

Findings

Entanglement Page curves are bounded by initial state participation entropies.

In the thermodynamic limit, Page curves for different circuit models coincide.

Classical permutation features significantly influence entanglement dynamics.

Abstract

The characterization of ensembles of many-qubit random states and their realization via quantum circuits are crucial tasks in quantum-information theory. In this work, we study the ensembles generated by quantum circuits that randomly permute the computational basis, thus acting classically on the corresponding states. We focus on the averaged entanglement and present two main results. First, we derive generically tight upper bounds on the entanglement that can be generated by applying permutation circuits to arbitrary initial states. We show that the late-time ``entanglement Page curves'' are bounded in terms of the initial state participation entropies and its overlap with the ``maximally antilocalized'' state. Second, comparing the averaged R\'enyi-$2$ entropies generated by $(i)$ an infinitely deep random circuit of two-qubit gates and $(ii)$ global random permutations, we show that the two quantities are different for finite $N$ but the corresponding Page curves coincide in the thermodynamic limit. We also discuss how these conclusions are modified by additional random phases or considering circuits of $k$-local gates with $k\geq 3$. Our results are exact and highlight the implications of classical features on entanglement generation in many-body systems.