Local quenches in chaotic quantum systems produce a Renyi-index-tuned hierarchy of entanglement transitions, with S_alpha>1 obeying area law while S_alpha<=1 is volume-law, carried by an O(1)-dimensional dominant Schmidt sector that itself exhibits similar transitions at lower critical indices.
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Higher moments of entanglement entropy distribution in hybrid quantum circuits distinguish measurement-induced phases and are captured by a phenomenological model for area-law combined with directed polymer description for volume-law.
Measurement-only circuits realize gapless SPT phases with nontrivial edge states at criticality, including symmetry-enriched percolation in Ising models and persistent Z4 gSPT phases mapped to Majorana loop models.
LC-inequivalent graph-state blocks in random Clifford circuits yield distinct entanglement velocities v_E and butterfly velocities v_B, correlated with internal entanglement distribution and graph connectivity.
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Gapless Symmetry-Protected Topological States in Measurement-Only Circuits
Measurement-only circuits realize gapless SPT phases with nontrivial edge states at criticality, including symmetry-enriched percolation in Ising models and persistent Z4 gSPT phases mapped to Majorana loop models.