Knill--Laflamme condition

A code projector $P$ corrects errors $\{E_a\}$ exactly when $P{E}_a^{†}{E}_{b}P=C_{ab}P$ for every pair.

If a common recovery quantum channel with Kraus operators $R_{\mu }$ restores every pure code state after each error $E_a$, then each vector $R_{\mu }{E}_a|\psi ⟩$ is parallel to $|\psi ⟩$. Linearity on basis vectors and their pairwise superpositions forces $R_{\mu }{E}_{a}P=c_{\mu a}P$, with the scalar independent of the logical state. Insert ${\sum }_{\mu }{R}_{\mu }^{†}{R}_{\mu }=I$ between two error operators to obtain the displayed Knill--Laflamme condition. This proves necessity without presuming an orthogonal syndrome description of recovery.

The matrix $C$ in the Knill--Laflamme condition is positive semidefinite, since $v^{†}Cv$ is the squared norm of ${\sum }_a{v}_a{E}_a|\psi ⟩$ for any normalized code state. Diagonalizing it gives linear combinations $F_j$ with $P{F}_j^{†}{F}_{l}P=c_j{\delta }_{jl}P$. For $c_j>0$, $V_j=F_{j}P/\sqrt{c_j}$ is an isometry of the code into an error-image subspace, and these subspaces are orthogonal. Measuring their projectors and applying $V_j^{†}$ 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.