Maximality of Quantum Error-Correcting Code Spaces

Quantum error-correcting is used in quantum computing to protect quantum information from errors due to decoherence and other quantum noises. An important task in quantum error-correcting is, for a given quantum channel, to find an error-correcting code space (ECCS) that contains more correctable information. In this paper, we introduce first the concept of a maximally error-correcting code space (MECCS) of a quantum channel and prove that any ECCS of a quantum channel must be contained in an MECCS of that channel. We also prove that every quantum channel has always an MECCS. Moreover, we prove a sufficient and necessary condition for the direct sum of two orthogonal ECCSs of the same channel to be an ECCS of that channel and therefore obtain a characterization of an MECCS. Lastly, we discuss the maximality of ECCSs of a quantum channel by introducing the concept of the maximal dimension C(𝓔)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C(\mathcal {E})$\end{document} of ECCSs of a quantum channel 𝓔\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {E}$\end{document}. A necessary and sufficient condition for C(𝓔)=k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C(\mathcal {E})=k$\end{document} is established by means of the rank-k numerical range of an operator.