Knill–Laflamme error-correction conditions

Let P project onto a code subspace and let the specified error operators be $E_a$. For each pair, compress $E_a^\dagger E_b$ into the code. Exact correctability requires this compression to be a scalar times P, with the scalar allowed to depend on the error labels but not on the logical state. The resulting matrix C is positive semidefinite. It records error-sector overlaps while leaving the encoded degree of freedom unchanged.

In an orthonormal logical basis, the condition contains two separate requirements. Off-diagonal logical matrix elements vanish, preventing distinct logical states from becoming confused. Diagonal elements are equal across the basis, preventing error probabilities or overlaps from distinguishing logical inputs. Both are needed. A procedure that successfully restores classical labels while their environmental records differ can erase the coherence of a superposition, even though measurements in the original logical basis look correct.

A unitary change of error basis diagonalizes C. After rescaling its nonzero eigenvalues, the transformed errors carry the code into mutually orthogonal subspaces. A syndrome measurement can distinguish these subspaces, and a conditional inverse can return each to the code. Crucially, it need not distinguish the logical vectors within any one subspace. Zero eigenvalues correspond to combinations that vanish on the code and require no separate recovery branch.

Orthogonal error subspaces are convenient but not necessary in the original error labeling. Two different physical error operators may act identically on every code state, making their distinction irrelevant. This is degeneracy. A recovery only has to identify enough information to undo the logical effect, rather than reconstruct a unique microscopic history. Confusing degeneracy with failure leads to overly strong demands on a syndrome and to incorrect classical sphere-counting arguments for quantum codes.

For the bit-flip repetition code, each single X takes both logical basis states into a new syndrome sector. Products of two different single X errors also take the code outside itself, so the relevant off-diagonal error overlaps vanish. In contrast, one Z preserves the code but has opposite signs on its logical basis states. Its compressed action is a logical operator, not a scalar. Adding unknown Z to the identity error set therefore violates the criterion.

The criterion is conditional on which operators are included. A known unitary can always be inverted, whereas an unknown choice between identity and a logical Z cannot be perfectly reversed without additional information. Approximate correction replaces exact identities by controlled error bounds and needs a chosen distance or fidelity measure. Neither a small violation nor a successful example establishes a universal hardware guarantee. Check the modeled Kraus span, inaccessible records, and recovery resources before interpreting the result physically.