Defect Relative Entropy

We introduce the concept of $\mathit{\text{defect}}\mathit{\text{relative}}\mathit{\text{entropy}}$ as a measure of distinguishability within the space of defects. We compute the defect relative entropy for conformal/topological defects, deriving a universal formula in conformal field theories (CFTs) on a circle. This formula reduces to the Kullback-Leibler divergence. Furthermore, we provide a detailed expression of the defect relative entropy for diagonal CFTs with specific topological defect choices, utilizing the theory’s modular $\mathcal{S}$ matrix. We also present a general formula for the $\mathit{\text{defect}}\mathit{\text{sandwiched}}\mathit{\text{R}}\acute{\mathit{\text{e}}}\mathit{\text{nyi}}\mathit{\text{relative}}\mathit{\text{entropy}}$ and the $\mathit{\text{defect}}\mathit{\text{fidelity}}$. Through explicit calculations in specific models, including the Ising model, the tricritical Ising model, and the $\widehat{su}(2{)}_k$ WZW model, we have made an intriguing finding: zero defect relative entropy between reduced density matrices associated with certain topological defect. Notably, we introduce the concept of the $\mathit{\text{defect}}\mathit{\text{relative}}\mathit{\text{sector}}$, representing the set of topological defects with zero defect relative entropy.