  In this work we investigate the conditions for quantum back action 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. Our
framework explicitly uses a model of unitary evolution which couples 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 them. We identify the possible replica symmetries an
experiment can possess and discuss 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 show how the entanglement transition can be understood as
the spontaneously 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 the brickwork quantum-enhanced
experiment introduced in an accompanying Letter [1] in the case where the
unitaries are chosen from the Haar measure, and identify a distinct
universality from 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.
