Defect Relative Entropy

We introduce the concept of defect relative 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 defect sandwiched Rényi relative entropy and the defect 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 defect relative sector, representing the set of topological defects with zero defect relative entropy.