Generalizing measurement-induced phase transitions to information exchange symmetry breaking

Probing a quantum system disrupts its state in a phenomenon known as backaction. In this work we investigate the conditions for this disruption to result in a phase transition in the information dynamics of a monitored system. We introduce a framework that captures a wide range of experiments encompassing probes comprised of projective measurements and probes which more generally transfer quantum information from the system to a quantum computer. The latter case is relevant to the recently introduced quantum-enhanced experiments in which quantum postprocessing is performed on a quantum computer and which can offer an exponential sampling advantage over projective measurements and classical postprocessing. Our framework explicitly considers the effects of an environment using a model of unitary evolution which couples the system, apparatus, and environment. Information dynamics is investigated using the R\'enyi and von Neumann entropies of the evolving state, and we construct a replica theory for studying these quantities. We identify the possible replica symmetries an experiment can possess and discuss the meaning of their spontaneous symmetry breaking. In particular, we identify a minimum subgroup whose spontaneous symmetry breaking results in an entanglement transition. This symmetry is only possible when the information in the apparatus is as informative about the dynamics of the system as the information transferred to the environment. We call this requirement the information exchange symmetry and quantify it by a relationship between the entropies. We then introduce a generalized notion of spontaneous symmetry breaking such that the entanglement transition can be understood as the spontaneous breaking of the information exchange symmetry and without referring to the replica theory. Information exchange symmetry breaking is then shown to generalize the phenomenology of the measurement-induced phase transition (MIPT). We apply this theory to a quantum-enhanced experiment probing the unitary dynamics of a Haar brickwork quantum circuit, and we determine that the universality class of the critical point is distinct from that for the MIPT. This notion of information exchange symmetry breaking generalizes the MIPT and provides a framework for understanding the dynamics of quantum information in quantum-enhanced experiments.