Characterizing Quantum Codes via the Coefficients in Knill-Laflamme Conditions

Quantum error correction (QEC) is essential for protecting quantum information against noise, yet understanding the structure of the Knill-Laflamme (KL) coefficients $\lambda_{ij}$ from the condition $PE_i^\dagger E_j P = \lambda_{ij} P$ remains challenging, particularly for nonadditive codes. In this work, we introduce the signature vector $\vec{\lambda}(P)$, composed of the off-diagonal KL coefficients $\lambda_{ij}$, where each coefficient corresponds to equivalence classes of errors counted only once. We define its Euclidean norm $\lambda^*(P)$ as a scalar measure representing the total strength of error correlations within the code subspace defined by the projector $P$. We parameterize $P$ on a Stiefel manifold and formulate an optimization problem based on the KL conditions to systematically explore possible values of $\lambda^*$. Moreover, we show that, for $((n,K,d))$ codes, $\lambda^*$ is invariant under local unitary transformations. Applying our approach to the $((6, 2, 3))$ quantum code, we find that $\lambda^*_{\text{min}} = \sqrt{0.6}$ and $\lambda^*_{\text{max}} = 1$, with $\lambda^* = 1$ corresponding to a known degenerate stabilizer code. We construct continuous families of new nonadditive codes parameterized by vectors in $\mathbb{R}^5$, with $\lambda^*$ varying over the interval $[\sqrt{0.6}, 1]$. For the $((7, 2, 3))$ code, we identify $\lambda^*_{\text{min}} = 0$ (corresponding to the non-degenerate Steane code) and $\lambda^*_{\text{max}} = \sqrt{7}$ (corresponding to the permutation-invariant code by Pollatsek and Ruskai), and we demonstrate continuous paths connecting these extremes via cyclic codes characterized solely by $\lambda^*$. Our findings provide new insights into the structure of quantum codes, advance the theoretical foundations of QEC, and open new avenues for investigating intricate relationships between code subspaces and error correlations.