Robust measurement-induced phase transitions and observables in noisy dynamics

Measurement-induced phase transitions generally suffer from two problems: fragility to noise and reliance on exponentially-costly postselection to be observed. Here, we seek to address both of these issues in a (2+1)d measurement-only model inspired by the toric code. We find noise-robust measurement-induced transitions in topological negativity and in purification timescales. Although these transitions rely on postselection, we study the performance of linear cross-entropy (LXE) as an efficiently-computable observable. Without modification, the LXE fails to show a phase transition in the presence of noise. However, we show how measurement results can be decoded to yield a robust LXE in certain regions of our phase diagram and we explore the theoretical limitations on this decoding.