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arXiv:2601.21875

Defect Relative Entropy

Mostafa Ghasemi

hep-thcond-mat.stat-mechmath-phquant-phSubmitted Jan 29, 2026• Announced Jan 30, 2026• 0 citations
v1 · Jan 29, 2026v2 · Jun 22, 2026
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Abstract

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.

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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 S\mathcal{S}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 su^(2)k\widehat{su}(2)_{k}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.
The concept of distinguishability lies at the heart of quantum information theory. We introduce \textit{left-right relative entropy} as a quantitative measure of distinguishability within the space of boundary states in two-dimensional conformal field theories (CFTs). By tracing over either the left- or right-moving modes, we derive a universal formula for arbitrary regularized boundary states defined on a circle. Remarkably, the resulting quantity reduces to a Kullback--Leibler divergence, where the associated probability distribution is determined entirely by the modular S\mathcal{S}S-matrix and the boundary data. For diagonal CFTs, we obtain exact expressions for the left-right relative entropy in terms of modular data, and extend the framework to define \textit{Sandwiched Left-Right Rényi relative entropies} and \textit{left-right fidelity}. Applying this formalism to the Ising model, tricritical Ising model, and su^(2)k\widehat{su}(2)_ksu(2)k​ WZW model, we uncover a striking phenomenon: the left-right relative entropy between certain reduced boundary states vanishes even though the corresponding global boundary states are orthogonal. This observation motivates the introduction of \textit{relative entanglement sectors}, defined as equivalence classes of boundary states that are indistinguishable with respect to left-right relative entropy. These sectors transform as NIM-representations of global symmetries and exhibit level-dependent structures that mirror Z2\mathbb{Z}_2Z2​ 't Hooft anomalies. Our findings establish an unexpected bridge between quantum information measures, boundary conformal symmetry, and quantum anomaly constraints.
In this work, we compute the defect relative entropy between topological defects in the symmetric product orbifold CFT SymN(M)=M⊗N/SN\mathrm{Sym}^N(M) = M^{\otimes N}/S_NSymN(M)=M⊗N/SN​. Our analysis covers two distinct classes of defects: universal defects, which realize the Rep(SN)\mathrm{Rep}(S_N)Rep(SN​) non-invertible symmetry, and non-universal defects. We show that the defect relative entropy reduces to a Kullback--Leibler (KL) divergence. The resulting expression decomposes naturally into two contributions: one governed by characters of the symmetric group SNS_NSN​, and the other controlled by modular SSS-matrix elements of the seed RCFT. Remarkably, both sets of data appear as probability distributions, yielding an information-theoretic interpretation of permutation group data and modular data within the symmetric orbifold. The structure of the divergence depends sensitively on the defect class. For universal defects, only the permutation group data contributes; for maximally fractional defects, both permutation and modular data enter and together define the relevant probability distributions. This feature suggests that the maximally fractional defect can be understood as a kind of product of the RCFT defect and the symmetric orbifold defect.
Entanglement entropy (EE) contains signatures of many universal properties of conformal field theories (CFTs), especially in the presence of boundaries or defects. In particular, topological defects are interesting since they reflect internal symmetries of the CFT and have been extensively analyzed with field-theoretic techniques with striking predictions. So far, however, no lattice computation of EE has been available. Here, we present an ab initio analysis of EE for the Ising model in the presence of a topological defect. While the behavior of the EE depends, as expected, on the geometric arrangement of the subsystem with respect to the defect, we find that zero-energy modes give rise to crucial finite-size corrections. Importantly, contrary to the field-theory predictions, the universal subleading term in the EE when the defect lies at the edge of the subsystem arises entirely due to these zero-energy modes and is not directly related to the modular S matrix of the Ising CFT.
Entanglement through interfaces has attracted considerable attention in 2d conformal field theory (CFT). However, it is known that field-theoretic predictions based on the existing framework are in general incompatible with numerical results [1-4]. A new framework for entanglement through topological defects was recently proposed in [5]. It provides a general description of entanglement through topological defects and successfully reproduces the numerical results for the Ising model in all tested cases and regimes. The key insight is that the relevant quantum correlations are encoded in twisted states, allowing for the construction of the full reduced density matrix (RDM). In this work we pursue two objectives. First, we provide new examples by studying defects in the free boson CFT. Second, we extend the framework to topological interfaces connecting two, possibly distinct, CFTs. Of particular interest are interfaces relating dual theories. We show that the reduced density matrix for a duality interface is the projection of the vacuum reduced density matrix onto a single symmetry sector, closely paralleling the framework of symmetry resolution. Unlike symmetry resolution, however, the projection is imposed by the physical interface itself, demonstrating that duality interfaces reflect quantum correlations back into the entangling interval. We establish this mechanism for diagonal and non-diagonal rational CFTs as well as the free boson CFT. Relative entropy allows us to quantify the distinguishability of the duality interface RDM from the vacuum RDM.
In this paper we consider entanglement entropies in two-dimensional conformal field theories in the presence of topological interfaces. Tracing over one side of the interface, the leading term of the entropy remains unchanged. The interface however adds a subleading contribution, which can be interpreted as a relative (Kullback-Leibler) entropy with respect to the situation with no defect inserted. Reinterpreting boundaries as topological interfaces of a chiral half of the full theory, we rederive the left/right entanglement entropy in analogy with the interface case. We discuss WZW models and toroidal bosonic theories as examples.
Relative entropy is a measure of distinguishability for quantum states, and it plays a central role in quantum information theory. The family of Renyi entropies generalizes to Renyi relative entropies that include, as special cases, most entropy measures used in quantum information theory. We construct a Euclidean path-integral approach to Renyi relative entropies in conformal field theory, then compute the fidelity and the relative entropy of states in one spatial dimension at zero and finite temperature using a replica trick. In contrast to the entanglement entropy, the relative entropy is free of ultraviolet divergences, and is obtained as a limit of certain correlation functions. The relative entropy of two states provides an upper bound on their trace distance.