Magic transition in monitored free fermion dynamics

Entanglement and magic are fundamental resources in quantum information, yet the interplay between their non-local structures in many-body systems remains largely unexplored. Measures of magic, such as the Stabilizer Rényi Entropy (SRE), often exhibit extensive scaling that obscures the intricate non-local correlations analogous to those captured by entanglement. To over come this, we investigate hybrid free-fermion circuits hosting a measurement-induced phase transition (MIPT), where SRE can be computed efficiently for large, highly entangled states via a perfect sampling algorithm. We introduce the Stabilizer Mutual Information (SMI) as a tool to dissect the non-local character of magic by filtering out extensive, short-range contributions. Our numerical simulations reveal that the bipartite SMI undergoes a phase transition that coincides precisely with the known entanglement transition. In the critical phase, characterized by logarithmic entanglement, the SMI also scales logarithmically with system size, indicating delocalized magic. In contrast, it saturates to a constant in the area-law phase, signifying that magic becomes localized. Additionally, we explore the dynamics of SRE. While the total SRE becomes extensive in O(1) time, we find that in the critical phase, the relaxation time to the steady-state value is parameterically longer than that in generic random circuits. The relaxation follows a universal form, with a relaxation time that grows linearly with the system size, providing further evidence for the critical nature of the phase.