Defect Relative Entropy

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

💡 Research Summary

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The paper introduces a new information‑theoretic quantity, the defect relative entropy (DRE), to quantify how distinguishable two defects (or interfaces) are within a quantum field theory. While entanglement entropy has been widely used to probe quantum correlations, it suffers from ultraviolet divergences and ambiguities in gauge theories. Relative entropy, by contrast, is UV‑finite, monotonic under completely positive trace‑preserving maps, and directly measures the distinguishability of two quantum states. The authors apply this concept to the reduced density matrices obtained by tracing out the complement of a spatial interval in a two‑dimensional conformal field theory (CFT) that contains a defect line.

The central technical achievement is the derivation of a universal formula for the DRE of topological defects on a circle. Starting from two arbitrary topological defects (I_{K}) and (I_{K’}), the authors employ the replica trick: the (n)-th replica partition function (Z_{n}(K,K’)) is expressed as a torus partition function with (2n) insertions of the defect operators. By using the commutation relations of topological defects with the Virasoro generators, the partition functions reduce to sums over characters weighted by the defect coefficients (d_{K}^{i}). After a modular S‑transformation and taking the thermodynamic limit (\ell/\epsilon\gg1), the DRE simplifies to a Kullback–Leibler (KL) divergence between two probability distributions: \