Quantum Coding Transitions in Unobserved Measurement-Driven Dynamics

While many properties of measurement-induced phase transitions(MIPT) are now understood, what defines the phases remains an outstanding problem. We hypothesize that these phases are determined by coding properties at short times and ergodicity at long times and seek MIPT-like phase transitions in unobserved dynamics. We focus on three common MIPT models: the monitored random unitary model(MRUM), projective transverse field Ising model(PTFIM), and projective honeycomb Kitaev model(PKHM). In all three models, commutative local quantum channels capture the unobserved dynamics and, viewed as communication through time, exhibit coding transitions at short times distinct from equilibrium properties with codes containing fewer logical qubits persisting to longer times. This result is conclusive for the PTFIM and PKHM because their unobserved quantum channels are degradable. For MRUM, we compute only the single-use quantum capacity, leaving the rich problem of $m$ uses unexplored, where, ideally, $m\to\infty$. For MRUM, we also study different Kraus operator-generated trajectory dynamics common in MIPT studies, specifically using Pauli, Clifford, and Haar's two-qubit unitaries, and find all three have different phase diagrams. Our results suggest that unobserved dynamics and different Kraus bases of observed dynamics all exhibit coding transitions at short times and ergodicity at long times and that post-selection is not an essential characteristic of a phase.