Characterizing the Structure of Preserved Information in Quantum Processes

We introduce a general operational characterization of information-preserving structures—encompassing noiseless subsystems, decoherence-free subspaces, pointer bases, and error-correcting codes—by demonstrating that they are isometric to fixed points of unital quantum processes. Using this, we show that every information-preserving structure is a matrix algebra. We further establish a structure theorem for the fixed states and observables of an arbitrary process, which unifies the Schrödinger and Heisenberg pictures, places restrictions on physically allowed kinds of information, and provides an efficient algorithm for finding all noiseless and unitarily noiseless subsystems of the process.