Measurement-Induced Phase Transition

Random quantum circuits alternate unitary gates (generating entanglement) with projective measurements (destroying it). At a critical measurement rate p_c, a phase transition occurs: from volume-law entanglement (quantum error-correcting) to area-law (product-like states).

p = 15%
L = 16
depth = 20
VOLUME LAW PHASE — Quantum error correcting code
Volume law: S ~ L (quantum information hidden from local measurements)
Area law: S ~ const (product-like, information revealed by measurements)
Critical p_c ≈ 16% (Haar random) | CFT description at transition with c~0.5