Partial post-selected measurements: Unveiling measurement-induced transitions trajectory by trajectory

Abstract

Measurement-induced Phase Transitions (MiPTs) emerge from the interplay between competing local quantum measurements and unitary scrambling dynamics. While monitored quantum trajectories are inherently stochastic, post-selecting specific detector readouts leads to dynamics governed by non-Hermitian Hamiltonians, revealing distinct universal characteristics of MiPTs.

Here, we contrast the quantum dynamics of individual post-selected trajectories with their collective statistics behavior. We introduce a novel partially post-selected stochastic Schrödinger equation that enables the study of controlled subsets of quantum trajectories. Applying this formalism to a Gaussian Majorana fermions model, we employ a two-replica approach combined with renormalization group (RG) techniques to demonstrate that non-Hermitian MiPT universality persists even under limited stochasticity. Notably, we discover that the transition to MiPT occurs at a finite partial post-selection threshold. Our findings establish a framework for leveraging non-Hermitian dynamics to investigate monitored quantum systems while addressing fundamental challenges in post-selection procedures.