Quantum stabilizer codes and classical linear codes
Richard Cleve · Physical Review A · 1997
We show that within any quantum stabilizer code there lurks a classical binary linear code with similar error-correcting capabilities. Using this result---which applies to degenerate as well as nondegenerate codes---previously established necessary conditions for classical linear codes can be easily translated into necessary conditions for quantum stabilizer codes. Examples of specific consequences are as follows: for a quantum channel subject to a \ensuremath{\delta} fraction of errors, the best asymptotic capacity attainable by any stabilizer code cannot exceed H(+$\sqrt{2\ensuremath{\delta}(1\ensuremath{-}2\ensuremath{\delta})}$); and, for the depolarizing channel with fidelity parameter \ensuremath{\delta}, the best asymptotic capacity attainable by any stabilizer code cannot exceed 1-H(\ensuremath{\delta}).