Metrological code[1] 

Two-dimensional subspace of a Hilbert space whose basis states satisfy only a part of the Knill-Laflamme conditions. The satisfied part of the conditions ensures that the code can be used for local parameter estimation.

Letting \(\Pi = U U^\dagger\) be the codespace projector for encoding isometry \(U\) and projecting a pair of errors \(E_i,E_j\) from an error set \(\cal E\) into the two-dimensional codespace yields \begin{align} \Pi E_{i}^{^{\dagger}}E_{j}\Pi=c_{ij}\,\Pi+x_{ij}\overline{X}+y_{ij}\overline{Y}+z_{ij}\overline{Z} \tag*{(1)}\end{align} with error-matrix element \(c_{ij}\) and logical-error coefficients \begin{align} \left\{ x,y,z\right\} _{ij}={\textstyle \frac{1}{2}}\text{Tr}\left(\left\{ \overline{X},\overline{Y},\overline{Z}\right\} E_{i}^{^{\dagger}}E_{j}\right)~. \tag*{(2)}\end{align} If all three logical-error coefficients are zero, then the Knill-Laflamme conditions are satisfied, and the code is a QECC. If only one of the three coefficients is zero, then the code is the more general metrological code.