= Constructive recovery from the Knill-Laflamme condition
{title2=$PE_a^\dagger E_bP=C_{ab}P$}
The matrix $C$ in the <Knill--Laflamme condition> is positive semidefinite, since $v^\dagger Cv$ is the squared norm of $\sum_av_aE_a|\psi\rangle$ for any normalized code state. Diagonalizing it gives linear combinations $F_j$ with $PF_j^\dagger F_lP=c_j\delta_{jl}P$. For $c_j>0$, $V_j=F_jP/\sqrt{c_j}$ is an isometry of the code into an error-image subspace, and these subspaces are orthogonal. Measuring their projectors and applying $V_j^\dagger$ recovers the logical state. Complete the operation arbitrarily on the orthogonal complement to obtain a <quantum channel>. This construction corrects every noise map whose <Kraus operators> lie in the specified error span, including coherent superpositions of error operators.
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