Measurement-Induced Phase Transitions in Many-Body Localized and Integrable Systems

Recent works have shown the competition between scrambling unitary dynamics and local projective measurements can lead to a phase transition in the dynamics of entanglement entropy. In integrable systems, the extensive number of conserved quantities strongly constrains the information scrambling. Here, we show that such systems can still exhibit an entanglement phase transition given an appropriate choice of measurement basis. We analyze a toy model of many-body localized systems as a paradigmatic example. If the observables being measured are not scrambled in unitary evolution, the growth of entanglement is prohibited by any finite rate of measurements. In contrast, if measured observables are scrambled, the unitary evolution can hide and protect quantum correlations from measurements, leading to a phase transition at a nonvanishing measurement rate. The phase transition in other integrable systems, such as free fermionic and Bethe-ansatz solvable systems are also explored. Our results further corroborate the understanding of the phase transition in terms of quantum error correcting properties of the scrambling dynamics.