Measurement-induced phase transition in space

Research Question and Approach

Traditional studies of measurement-induced phase transitions (MIPTs) typically rely on uniform measurement probabilities across an entire quantum system, requiring multiple independent simulations to reconstruct the phase transition. This paper investigates whether a spatial gradient in measurement probability can encode the entire transition—from volume-law entanglement to area-law entanglement—within a single, spatially inhomogeneous steady state. The authors employ a (1+1)D monitored Clifford circuit where the measurement probability $p(x)$ varies linearly across the chain, effectively creating a spatial map of the phase diagram.

The researchers show that the spatial MIPT is governed by a scaling form structurally analogous to finite-time scaling in temporally driven systems. By scanning entanglement observables across the spatial profile, they identify a critical point $p(x) = p_c$ that acts as a spatial cut. Unlike temporal protocols, which are limited by critical slowing down (Kibble-Zurek dynamics), the spatial protocol is constrained by the physical bounds of the measurement probability ($0 \le p \le 1$). This creates a finite linear window that acts as a geometric cutoff, allowing the authors to extract the correlation-length exponent $\nu$ directly from the asymptotic behavior of the entanglement entropy on both sides of the critical point.

This approach provides a highly efficient, resource-saving method for probing quantum criticality. By engineering a single steady state that contains the entire transition, researchers can observe the coexistence of different entanglement phases simultaneously. This spatial realization offers a new, controlled route for exploring measurement-induced phenomena in near-term quantum simulators, potentially reducing the number of experimental runs required to characterize complex quantum phases.