Characterizing Quantum Codes via the Coefficients in Knill-Laflamme Condition

Quantum error correction (QEC) is vital for protecting quantum information from noise. However, understanding the structure of Knill-Laflamme coefficients, which describe how errors impact the code, is challenging, especially for nonadditive codes. In this work, we introduce the signature vector, composed of unique off-diagonal Knill-Laflamme coefficients representing distinct error classes. We define its Euclidean norm, λ*, as a measure of the total strength of error correlations within the code subspace. By parameterizing the projector on a Stiefel manifold and formulating an optimization problem based on the Knill-Laflamme conditions, we systematically explore possible λ* values. We show that for ((n, K, d)) codes, λ* is invariant under local unitary transformations. Applying our approach to the ((6, 2, 3)) quantum code, we find λ* ranges from sqrt(0.6) to 1, with the maximum corresponding to a known stabilizer code. We construct continuous families of new nonadditive codes parameterized by five-dimensional vectors, with λ* varying between sqrt(0.6) and 1. For the ((7, 2, 3)) code, we identify λ* ranging from 0 (the Steane code) to sqrt(7) (the Pollatsek-Ruskai code) and demonstrate paths connecting these extremes through cyclic codes characterized by λ*. Our findings provide new insights into quantum code structures, enhance the theoretical foundations of QEC, and open new avenues for exploring relationships between code subspaces and error correlations.