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Résumé |
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. Generally speaking, this result states that
the quantum correlations generated by classical circuits are bounded in terms of some quantum property of
the initial state (namely, the degree to which it can be written as a superposition of classical states). 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.
We certify the emergent quantum randomness using the frame potential and establish a mapping between its
behavior and the statistical mechanics of a domain wall particle model. In both architectures, the effect of
quantum measurements induces a nontrivial confinement mechanism, where domain walls are either trapped
by an external potential or bound in pairs to form meson-like excitations. Our results, supported by both exact
analytical calculations and numerical simulations, suggest that confinement is a general mechanism
underlying random state generation in broader settings with local measurements, including quantum circuits
and chaotic dynamics. |
