Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2601.21875

Defect Relative Entropy

Mostafa Ghasemi

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

Abstract

Distinguishability is central to quantum information theory, but a quantitative measure for distinguishing topological defects--realizations of generalized symmetries in quantum field theory (QFT)--has been lacking. We introduce the notion of the \textit{defect relative entropy} to fill this gap for topological defects in two-dimensional conformal field theories (CFTs). For defects on a circle, we derive a universal formula that reduces defect relative entropy to a Kullback--Leibler divergence determined entirely by the modular $\mathcal{S}$-matrix and defect coefficients. Thus, the algebraic data governing modular transformations also determines defect distinguishability. A striking consequence is that certain distinct defects can have vanishing relative entropy when restricted to one side, implying that an observer confined to that side cannot distinguish them. This gives rise to information-theoretic equivalence classes of defects, which we term \textit{defect relative sectors}. We further introduce the sandwiched defect Rényi relative entropy and defect fidelity, derive general formulas for these quantities. Explicit calculations in the Ising model, tricritical Ising model, and $\widehat{su}(2)_k$ WZW models illustrate our results.

Comments
5 pages+Supplemental Material

Related papers

Left-Right Relative Entropy

Mostafa Ghasemi

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 $\mathcal{S}$-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 $\widehat{su}(2)_k$ 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 $\mathbb{Z}_2$ 't Hooft anomalies. Our findings establish an unexpected bridge between quantum information measures, boundary conformal symmetry, and quantum anomaly constraints.

arXiv:2411.09406•Nov 14, 2024•0 citations

Defect relative entropy in symmetric orbifold CFTs

Mostafa Ghasemi

In this work, we compute the defect relative entropy between topological defects in the symmetric product orbifold CFT $\mathrm{Sym}^N(M) = M^{\otimes N}/S_N$. Our analysis covers two distinct classes of defects: universal defects, which realize the $\mathrm{Rep}(S_N)$ 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 $S_N$, and the other controlled by modular $S$-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.

hep-thCross-listedarXiv:2602.13782•Feb 17, 2026•0 citations

Entanglement Entropy in the Ising Model with Topological Defects

Ananda Roy, Hubert Saleur

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.

arXiv:2111.04534•Physical Review Letters•Mar 2, 2022•51 citations

Entanglement in Presence of Topological Interfaces and Dualities

Christian Northe, Riccardo Poletti, Paolo Rossi

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.

hep-thCross-listedarXiv:2607.22451•Jul 27, 2026•0 citations

Entanglement and topological interfaces

E. Brehm, Ilka Brunner A, D. Jaud +1 more

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.

arXiv:1512.05945•Fortschritte der Physik•Apr 13, 2016•31 citations

Relative Entropies in Conformal Field Theory

Nima Lashkari

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.

arXiv:1404.3216•Physical Review Letters•Jul 29, 2014•99 citations
Distinguishability is central to quantum information theory, but a quantitative measure for distinguishing topological defects--realizations of generalized symmetries in quantum field theory (QFT)--has been lacking. We introduce the notion of the \textit{defect relative entropy} to fill this gap for topological defects in two-dimensional conformal field theories (CFTs). For defects on a circle, we derive a universal formula that reduces defect relative entropy to a Kullback--Leibler divergence determined entirely by the modular S\mathcal{S}S-matrix and defect coefficients. Thus, the algebraic data governing modular transformations also determines defect distinguishability. A striking consequence is that certain distinct defects can have vanishing relative entropy when restricted to one side, implying that an observer confined to that side cannot distinguish them. This gives rise to information-theoretic equivalence classes of defects, which we term \textit{defect relative sectors}. We further introduce the sandwiched defect Rényi relative entropy and defect fidelity, derive general formulas for these quantities. Explicit calculations in the Ising model, tricritical Ising model, and su^(2)k\widehat{su}(2)_ksu(2)k​ WZW models illustrate our results.
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.