Self-Organized Error Correction in Random Unitary Circuits with Measurement*

Random measurements have been shown to induce a phase transition in 1D quantum system evolving under chaotic local unitary dynamics. In this talk, we will the subthermal volume law phase and its connection to quantum error-correcting codes. We study the mutual information between a qudit inside a region and the exterior and identifying a power law decay with distance from the region's boundary with power 3/2. It implies that measurement deep inside the region will have negligible effect on its entanglement. We also find a universal, subleading logarithmic contribution to the volume law entanglement entropy, which is intimately related to the first observation in our formalism. Finally, we adopt the quantum Hamming bound to obtain a bound on the critical measurement strength as a function of the qudit dimension. The bound is saturated at infintie qudit dimension and provides a reasonable estimate for the qubit transition.