Irreversibility and Quantum Information Flow under Global & Local Gauge Symmetries ^†

What is the structure of general quantum processes on composite systems that respect a global or local symmetry principle? How does the irreversible use of quantum resources behave under such symmetry principles? Here we develop an information-theoretic framework to address these questions and show that every symmetric quantum process on a system has an essentially unique decomposition in terms of the flow of symmetry-breaking degrees of freedom between each subsystem and its environment. The decomposition has a natural causal structure that can be represented diagrammatically and makes explicit gauge degrees of freedom between subsystems. Our framework also provides a novel quantum information perspective on lattice gauge theories and a method to gauge general quantum processes beyond Lagrangian formulations. This procedure admits a simple resource-theoretic interpretation, and thus offers a natural context in which features such as information flow and entanglement in gauge theories could be studied. The framework also provides a flexible toolkit with which to analyse the structure of general quantum processes. As an application, we make use of a ‘polar decomposition’ for quantum processes to discuss incompatibility in the use of quantum resources and to provide a novel perspective in terms of the geometry induced on the orbit of a local process under a symmetry action.