Editing Quantum error correction (section)

==General codes==
In general, a ''quantum code'' for a [[quantum channel]] <math>\mathcal{E}</math> is a subspace <math>\mathcal{C} \subseteq \mathcal{H}</math>, where <math>\mathcal{H}</math> is the state Hilbert space, such that there exists another quantum channel <math>\mathcal{R}</math> with
<math display="block"> (\mathcal{R} \circ \mathcal{E})(\rho) = \rho \quad \forall \rho = P_{\mathcal{C}}\rho P_{\mathcal{C}},</math>
where <math>P_{\mathcal{C}}</math> is the [[orthogonal projection]] onto <math>\mathcal{C}</math>. Here <math>\mathcal{R}</math> is known as the ''correction operation''.

A ''non-degenerate code'' is one for which different elements of the set of correctable errors produce linearly independent results when applied to elements of the code. If distinct of the set of correctable errors produce orthogonal results, the code is considered ''pure''.<ref>{{cite journal |first1=A. R. |last1=Calderbank |first2=E. M. |last2=Rains |first3=P. W. |last3=Shor |first4=N. J. A. |last4=Sloane |title=Quantum Error Correction via Codes over GF(4) |journal=IEEE Transactions on Information Theory |volume=44 |issue=4 |year=1998 |pages=1369–1387 |doi=10.1109/18.681315 |arxiv=quant-ph/9608006 |s2cid=1215697 }}</ref>