Complete understanding of entanglement behavior across measurement rates in hybrid quantum circuits

Establish a complete theoretical description of the entanglement entropy behavior across the full range of measurement probabilities in one-dimensional hybrid quantum circuits composed of random unitary gates and local projective measurements, with particular emphasis on the area-law phase where no exact characterization currently exists.

Background

Hybrid quantum circuits, which interleave random unitary gates with local measurements, exhibit a measurement-induced phase transition in entanglement scaling between volume-law and area-law phases. While mappings to statistical-mechanics models have provided significant insights—most notably an exact directed polymer in a random environment (DPRE) description in the volume-law phase—there is no comparably complete framework that captures entanglement behavior across the entire range of measurement probabilities.

This paper demonstrates that higher moments of the entanglement entropy distribution (e.g., index of dispersion and skewness) offer robust diagnostics of the transition and proposes phenomenological and coarse-grained models for the area-law regime. Despite these advances, a comprehensive and unified theoretical understanding, especially in the area-law phase, remains unresolved.