Phases of information encoding in many-body quantum states

Intermediate-scale quantum devices operate controllably and can maintain quantum entanglement in systems of up to a few hundred qubits. From a many-body perspective, these devices exhibit a novel interplay of entangling unitary evolution, measurements, and decoherence, which is expected to give rise to new emergent phenomena. In this dissertation, we study the collective phenomena associated with information encoding and focus on two broad classes of systems: (i) monitored quantum circuits that consist of unitary evolution and measurements; (ii) topologically ordered states subject to local decoherence. Monitored random quantum circuits have been shown to undergo a measurement-induced phase transition in the steady state when increasing the measurement rate. In the first part of the dissertation, we develop a theoretical framework that maps the quantum information dynamics in monitored circuits to equilibrium statistical mechanics models. We show that the measurement-induced transition can be formulated as a transition in the capacity of the circuit to encode quantum information and further as symmetry-breaking transitions in statistical-mechanical models. Within this framework, we also identify new phases of information flow in monitored circuits, both when symmetry is imposed on circuit elements and when the circuit is at finite times.Topologically ordered states in quantum codes can store quantum information in the presence of local decoherence up to a finite threshold. In the second part of this dissertation, we study the effect of decoherence on general quantum ground states with topological order. We develop a theoretical framework based on effective field theory to identify the possible phases induced by decoherence and characterize their capacity to encode quantum information. We further propose information-theoretical quantities to define topological order in the ensuing decoherence-induced mixed states