  A formalism for quantum error correction based on operator algebras was
introduced in [1] via consideration of the Heisenberg picture for quantum
dynamics. The resulting theory allows for the correction of hybrid
quantum-classical information and does not require an encoded state to be
entirely in one of the corresponding subspaces or subsystems. Here, we provide
detailed proofs for the results of [1], derive a number of new results, and we
elucidate key points with expanded discussions. We also present several
examples and indicate how the theory can be extended to operator spaces and
general positive operator-valued measures.
