Quantum information encoding, protection, and correction from trace-norm isometries

We introduce the notion of trace-norm isometric encoding and explore its implications for passive and active methods for protecting quantum information against errors. Beside providing an operational foundation to the “subsystems principle” [E. Knill, Phys. Rev. A 74, 042301 (2006)] for faithfully realizing quantum information in physical systems, our approach allows additional explicit connections between noiseless, protectable, and correctable quantum codes to be identified. Robustness properties of isometric encodings against imperfect initialization and/or deviations from the intended error models are also analyzed.

Figure 1

Pictorial representation of trace-norm isometries in quantum information encoding, correction, and protection. Here ${\tilde{\mathcal{C}}}_Q$ denotes a code isomorphic to ${\mathcal{C}}_Q$, whereas the (state-)decoding map returns information from the physical back to the abstract level.