Knill–Laflamme conditions

Remarks

Proof idea: if a recovery $\mathcal{R}$ with Kraus operators $R_k$ exists, then $R_k E_a P = \lambda_{ka} P$ for scalars $\lambda_{ka}$, and $P E_a^* E_b P = \sum_k \overline{\lambda_{ka}} \lambda_{kb} P$. Conversely, diagonalise $(c_{ab})$ by a unitary change of Kraus operators to get $F_a$ with $P F_a^* F_b P = d_a \delta_{ab} P$. Then the subspaces $F_a \mathcal{C}$ with $d_a > 0$ are mutually orthogonal copies of $\mathcal{C}$. The recovery measures which subspace holds the state and maps it back to $\mathcal{C}$ by the partial isometry $d_a^{-1/2} P F_a^*$.

Example: the three-qubit bit-flip code $\mathcal{C} = \operatorname{span}\set{\lvert 000 \rangle, \lvert 111 \rangle}$ and the Kraus operators $\sqrt{p_0} I$ and $\sqrt{p_i} X_i$ for $i = 1, 2, 3$. For $i \ne j$, $X_i X_j$ and $X_i$ map $\mathcal{C}$ to a space orthogonal to $\mathcal{C}$, so $P X_i X_j P = 0$ and $P X_i P = 0$, while $P X_i X_i P = P$. So the conditions hold, with $(c_{ab})$ diagonal.

The conditions depend only on the linear span of the $E_a$. So a code that corrects every error in a set $\set{E_a}$ corrects every channel with Kraus operators in their span.

Going deeper

For a student

The conditions are the starting point of most constructions of quantum codes. A code has distance $d$ when the conditions hold for all Pauli errors of weight at most $d - 1$ in the role of $E_a^* E_b$. Then it corrects arbitrary errors on any $\lfloor (d-1)/2 \rfloor$ qubits. For stabilizer codes the conditions become a simple check on the commutation of Pauli operators with the stabilizer. When $(c_{ab})$ is not of full rank, the code is called degenerate: different errors act in the same way on it.