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Let $P$ be the orthogonal projector onto the quantum code $\mathcal{X}$, and choose an orthonormal basis $|i_L⟩$ for it. The necessary and sufficient Knill--Laflamme condition is that one matrix $C$, independent of the logical labels, satisfiesEquivalently, $⟨i_L|E_a^{†}{E}_b|j_L⟩=C_{ab}{\delta }_{ij}$. This makes the information in the error label independent of the encoded quantum state. It applies to the whole linear span of the errors, and hence to every noise quantum channel with Kraus operators in that span.

Here is a proof of both directions. For necessity of the Knill-Laflamme condition, let $R_{\mu }$ be Kraus operators of a common recovery, so ${\sum }_{\mu }{R}_{\mu }^{†}{R}_{\mu }=I$. Since recovery after $E_a$ must return the pure input state, every vector $R_{\mu }{E}_a|\psi ⟩$ is parallel to $|\psi ⟩$. A linear operator on the code which sends every vector to a multiple of itself is scalar: apply it to two basis vectors and their sum to force the two multipliers to agree. Therefore $R_{\mu }{E}_{a}P=c_{\mu a}P$ with no dependence on $|\psi ⟩$. It follows thatThis proves necessity without assuming in advance that the physical errors have orthogonal syndromes.

For sufficiency, $C$ is positive semidefinite: for any normalized code state and coefficients $v_a$, $v^{†}Cv=\Vert {\sum }_a{v}_a{E}_a|\psi ⟩{\Vert }^2\ge 0$. Diagonalize $C$ by a unitary change of the error basis to get operators $F_j$ with $P{F}_j^{†}{F}_{l}P=c_j{\delta }_{jl}P$. If $c_j>0$, $V_j=F_{j}P/\sqrt{c_j}$ is a linear isometry from the code to an error-image subspace, and distinct such images are orthogonal. If $c_j=0$, then $F_{j}P=0$. A projective measurement of these image subspaces followed by $V_j^{†}$ recovers every logical state. Complete the recovery on the unused orthogonal complement in any trace-preserving manner. For a coherent error ${\sum }_j{v}_j{F}_j$, the measurement branch $j$ contains $v_j\sqrt{c_j}|\psi ⟩$; the branch probability is independent of the logical state, and discarding the syndrome leaves the original state. This is constructive recovery from the Knill-Laflamme condition.

For the usual family of distinct unitary Pauli errors modulo scalar phase, nondegenerate quantum code correction means that different physical errors have orthogonal image subspaces. The precise condition isFor nonunitary errors, replace the unit diagonal by positive norm factors. In an arbitrary error basis the Gram matrix need not initially be diagonal; its positive-eigenvalue modes give the independent syndrome spaces just constructed. A kernel of $C$ exhibits a linear combination of errors acting as zero on the code. In particular two distinct physical errors with the same action on the code cannot be distinguished and give degenerate correction, as in part (c).

The physical space of $n$ qubits is ${\mathcal{H}}_n=({\mathbb{C}}^2{)}^{\otimes n}$. An error operator is a linear operator on this space, for example one Kraus operator of a noise quantum channel. The usual channel description assumes initially uncorrelated system and environment, with the environment subsequently ignored; more general initial correlations need not define a channel on arbitrary system inputs. Finite dimension and the tensor decomposition into identifiable qubits allow an exact Pauli expansion of a quantum error. No independence between errors on different qubits is assumed by this expansion.

The $4^n$ tensor products of $I,X,Y,Z$ form an orthogonal basis of the operator space under the Hilbert-Schmidt inner product. Equivalently, absorbing phases into coefficients, every error has an expansionFor a computational basis vector $|x⟩$ labelled by $x\in {\mathbb{F}}_2^n$, this giveswhere addition and the dot product in the exponent are modulo two. Thus $X$ changes a bit, $Z$ changes its phase, and $Y$ performs both up to a scalar phase. A general error is a coherent linear combination of such actions; it need not be a classical random choice of Pauli operators. A restriction to at most $t$ faulty qubits is an additional locality assumption, expressed by the span of Pauli operators of weight at most $t$.

The weight of a Pauli operator is the number of tensor factors different from $I$, ignoring global phase. Let $P$ be the orthogonal projector onto a $2^k$-dimensional code $\mathcal{X}$. Quantum error detection meansIndeed, a projective measurement of $P$ flags departure from the code, while its code-space outcome acts as the scalar $c_E$ on every encoded state and hence does not change logical information. Harmless scalar actions are included in this criterion. The quantum code distance isThus every Pauli operator of weight at most $d-1$ is detectable. The condition is linear in $E$, so it holds for any error in their span. This proves detection of arbitrary errors affecting at most $d-1$ qubits, including coherent errors.

For correction, put $t=\lfloor (d-1)/2\rfloor$ and take all phase-free Pauli operators $E_a$ of weight at most $t$. Their pairwise products satisfyThese are the Knill--Laflamme conditions. Here is their recovery construction, rather than merely using the criterion's name. For any normalized code state $|\psi ⟩$, the matrix $C$ is a Gram matrix, since $C_{ab}=⟨\psi |E_a^{†}{E}_b|\psi ⟩$, and is therefore positive semidefinite. Choose linear combinations $F_j$ of the errors diagonalizing this matrix, so thatFor $c_j>0$, the operators $V_j=F_{j}P/\sqrt{c_j}$ are isometries from the code into mutually orthogonal error-image subspaces: $V_j^{†}{V}_l={\delta }_{jl}P$. Measure the projectors $Q_j=V_j{V}_j^{†}$ and, on outcome $j$, apply $V_j^{†}$. Complete this recovery arbitrarily on the remaining orthogonal subspace to make a trace-preserving quantum channel.

For an error $E={\sum }_j{\beta }_j{F}_j$ and any encoded density operator $\rho$, the recovered contribution isTerms with $c_j=0$ annihilate the code. For a physical noise channel whose Kraus operators belong to this span, summing the displayed scalar over those operators gives one by trace preservation. The resulting recovered state is exactly $\rho$. This proves the constructive recovery from the Knill-Laflamme condition and the requested guaranteeThis argument allows degenerate codes; distinct physical errors may have identical actions on the code.

For the Shor code, define $|G_{\pm }⟩=(|000⟩\pm |111⟩)/\sqrt{2}$. Its two normalized basis codewords areSix stabilizer generators are $Z_1{Z}_2,Z_2{Z}_3,Z_4{Z}_5,Z_5{Z}_6,Z_7{Z}_8,Z_8{Z}_9$; the two phase checks areAll eight have eigenvalue $+1$ on both basis codewords. A phase error $Z_4$ commutes with the six $Z$ checks and anticommutes with both $g_7$ and $g_8$. Their projective measurements therefore returnThe two negative phase checks identify the middle three-qubit block. Apply the Pauli Z gate $Z_4$ to restore the state, since $Z_4^2=I$. This procedure acts identically on every logical superposition, and the checks reveal no logical amplitudes. The syndrome cannot distinguish $Z_4$ from $Z_5$ or $Z_6$, but all three have the same action on the code: their pairwise products lie in the stabilizer group. Correcting with any one of them is sufficient. Thus the degeneracy in the syndrome is harmless.