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:2112.09000 · Journal of High Energy Physics

Generalized entanglement entropies in two-dimensional conformal field theory

Sara Murciano, Pasquale Calabrese, Robert Konik

Published May 23, 2022• 29 citations
arXivHTMLPDF

Abstract

Abstract We introduce and study generalized Rényi entropies defined through the traces of products of TrB(| Ψi⟩⟨Ψj| ) where ∣Ψi⟩ are eigenstates of a two-dimensional conformal field theory (CFT). When ∣Ψi⟩ = ∣Ψj⟩ these objects reduce to the standard Rényi entropies of the eigenstates of the CFT. Exploiting the path integral formalism, we show that the second generalized Rényi entropies are equivalent to four point correlators. We then focus on a free bosonic theory for which the mode expansion of the fields allows us to develop an efficient strategy to compute the second generalized Rényi entropy for all eigenstates. As a byproduct, our approach also leads to new results for the standard Rényi and relative entropies involving arbitrary descendent states of the bosonic CFT.

DOI
10.1007/jhep05(2022)152

Related papers

Rényi entropy and geometry

Jeongseog Lee, Lauren McGough, Benjamin R. Safdi

Entanglement entropy in even-dimensional conformal field theories (CFTs) contains well-known universal terms arising from the conformal anomaly. R\'enyi entropies are natural generalizations of the entanglement entropy that are much less understood. Above two spacetime dimensions, the universal terms in the R\'enyi entropies are unknown for general entangling geometries. We conjecture a new structure in the dependence of the four-dimensional R\'enyi entropies on the intrinsic and extrinsic geometry of the entangling surface. We provide evidence for this conjecture by direct numerical computations in the free scalar and fermion field theories. The computation involves relating the four-dimensional free massless R\'enyi entropies across cylindrical entangling surfaces to corresponding three-dimensional massive R\'enyi entropies across circular entangling surfaces. Our numerical technique also allows us to directly probe other interesting aspects of three-dimensional R\'enyi entropy, including the massless renormalized R\'enyi entropy and calculable contributions to the perimeter law.

arXiv:1403.1580•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Jun 18, 2014•31 citations

Relative entanglement entropies in 1 + 1-dimensional conformal field theories

Paola Ruggiero, Pasquale Calabrese

We study the relative entanglement entropies of one interval between excited states of a 1+1 dimensional conformal field theory (CFT). To compute the relative entropy S(ρ 1∥ρ 0) between two given reduced density matrices ρ 1 and ρ 0 of a quantum field theory, we employ the replica trick which relies on the path integral representation of Tr(ρ 1 ρ 0 − 1 ) and define a set of Rényi relative entropies S n (ρ 1∥ρ 0). We compute these quantities for integer values of the parameter n and derive via the replica limit the relative entropy between excited states generated by primary fields of a free massless bosonic field. In particular, we provide the relative entanglement entropy of the state described by the primary operator i∂ϕ, both with respect to the ground state and to the state generated by chiral vertex operators. These predictions are tested against exact numerical calculations in the XX spin-chain finding perfect agreement.

arXiv:1612.00659•Journal of High Energy Physics•Feb 1, 2017•49 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

Entanglement in descendants

Barsha G. Chowdhury, Justin R. David

Abstract We study the single interval entanglement and relative entropies of conformal descendants in 2d CFT. Descendants contain non-trivial entanglement, though the entanglement entropy of the canonical primary in the free boson CFT contains no additional entanglement compared to the vacuum, we show that the entanglement entropy of the state created by its level one descendant is non-trivial and is identical to that of the U(1) current in this theory. We determine the first sub-leading corrections to the short interval expansion of the entanglement entropy of descendants in a general CFT from their four point function on the n-sheeted plane. We show that these corrections are determined by multiplying squares of appropriate dressing factors to the corresponding corrections of the primary. Relative entropy between descendants of the same primary is proportional to the square of the difference of their dressing factors. We apply our results to a class of descendants of generalized free fields and descendants of the vacuum and show that their dressing factors are universal.

arXiv:2108.00898•Journal of High Energy Physics•Feb 1, 2022•0 citations

Genus two partition functions and Rényi entropies of large c conformal field theories

Alexandre Belin, Christoph A. Keller, Ida G. Zadeh

Abstract We compute genus two partition functions in two-dimensional conformal field theories at large central charge, focusing on surfaces that give the third Rényi entropy of two intervals. We compute this for generalized free theories and for symmetric orbifolds, and compare it to the result in pure gravity. We find a new phase transition if the theory contains a light operator of dimension Δ ⩽ 0.19 . This means in particular that unlike the second Rényi entropy, the third one is no longer universal.

arXiv:1704.08250•Journal of Physics A Mathematical and Theoretical•Sep 4, 2017•51 citations

Thermal corrections to Rényi entropies for conformal field theories

Christopher P. Herzog, Jun Nian

We compute thermal corrections to Rényi entropies of d dimensional conformal field theories on spheres. Consider the nth Rényi entropy for a cap of opening angle 2θ on S d−1. From a Boltzmann sum decomposition and the operator-state correspondence, the leading correction is related to a certain two-point correlation function of the operator (not equal to the identity) with smallest scaling dimension. More specifically, via a conformal map, the correction can be expressed in terms of the two-point function on a certain conical space with opening angle 2πn. In the case of free conformal field theories, this two-point function can be computed explicitly using the method of images. We perform the computation for the conformally coupled scalar. From the n → 1 limit of our results, we extract the leading thermal correction to the entanglement entropy, reproducing results of arXiv:1407.1358.

arXiv:1411.6505•Journal of High Energy Physics•Jun 1, 2015•13 citations
Abstract We introduce and study generalized Rényi entropies defined through the traces of products of TrB(| Ψi⟩⟨Ψj| ) where ∣Ψi⟩ are eigenstates of a two-dimensional conformal field theory (CFT). When ∣Ψi⟩ = ∣Ψj⟩ these objects reduce to the standard Rényi entropies of the eigenstates of the CFT. Exploiting the path integral formalism, we show that the second generalized Rényi entropies are equivalent to four point correlators. We then focus on a free bosonic theory for which the mode expansion of the fields allows us to develop an efficient strategy to compute the second generalized Rényi entropy for all eigenstates. As a byproduct, our approach also leads to new results for the standard Rényi and relative entropies involving arbitrary descendent states of the bosonic CFT.
Entanglement entropy in even-dimensional conformal field theories (CFTs) contains well-known universal terms arising from the conformal anomaly. R\'enyi entropies are natural generalizations of the entanglement entropy that are much less understood. Above two spacetime dimensions, the universal terms in the R\'enyi entropies are unknown for general entangling geometries. We conjecture a new structure in the dependence of the four-dimensional R\'enyi entropies on the intrinsic and extrinsic geometry of the entangling surface. We provide evidence for this conjecture by direct numerical computations in the free scalar and fermion field theories. The computation involves relating the four-dimensional free massless R\'enyi entropies across cylindrical entangling surfaces to corresponding three-dimensional massive R\'enyi entropies across circular entangling surfaces. Our numerical technique also allows us to directly probe other interesting aspects of three-dimensional R\'enyi entropy, including the massless renormalized R\'enyi entropy and calculable contributions to the perimeter law.
We study the relative entanglement entropies of one interval between excited states of a 1+1 dimensional conformal field theory (CFT). To compute the relative entropy S(ρ 1∥ρ 0) between two given reduced density matrices ρ 1 and ρ 0 of a quantum field theory, we employ the replica trick which relies on the path integral representation of Tr(ρ 1 ρ 0 − 1 ) and define a set of Rényi relative entropies S n (ρ 1∥ρ 0). We compute these quantities for integer values of the parameter n and derive via the replica limit the relative entropy between excited states generated by primary fields of a free massless bosonic field. In particular, we provide the relative entanglement entropy of the state described by the primary operator i∂ϕ, both with respect to the ground state and to the state generated by chiral vertex operators. These predictions are tested against exact numerical calculations in the XX spin-chain finding perfect agreement.
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.
Abstract We study the single interval entanglement and relative entropies of conformal descendants in 2d CFT. Descendants contain non-trivial entanglement, though the entanglement entropy of the canonical primary in the free boson CFT contains no additional entanglement compared to the vacuum, we show that the entanglement entropy of the state created by its level one descendant is non-trivial and is identical to that of the U(1) current in this theory. We determine the first sub-leading corrections to the short interval expansion of the entanglement entropy of descendants in a general CFT from their four point function on the n-sheeted plane. We show that these corrections are determined by multiplying squares of appropriate dressing factors to the corresponding corrections of the primary. Relative entropy between descendants of the same primary is proportional to the square of the difference of their dressing factors. We apply our results to a class of descendants of generalized free fields and descendants of the vacuum and show that their dressing factors are universal.
Abstract We compute genus two partition functions in two-dimensional conformal field theories at large central charge, focusing on surfaces that give the third Rényi entropy of two intervals. We compute this for generalized free theories and for symmetric orbifolds, and compare it to the result in pure gravity. We find a new phase transition if the theory contains a light operator of dimension Δ ⩽ 0.19 . This means in particular that unlike the second Rényi entropy, the third one is no longer universal.
We compute thermal corrections to Rényi entropies of d dimensional conformal field theories on spheres. Consider the nth Rényi entropy for a cap of opening angle 2θ on S d−1. From a Boltzmann sum decomposition and the operator-state correspondence, the leading correction is related to a certain two-point correlation function of the operator (not equal to the identity) with smallest scaling dimension. More specifically, via a conformal map, the correction can be expressed in terms of the two-point function on a certain conical space with opening angle 2πn. In the case of free conformal field theories, this two-point function can be computed explicitly using the method of images. We perform the computation for the conformally coupled scalar. From the n → 1 limit of our results, we extract the leading thermal correction to the entanglement entropy, reproducing results of arXiv:1407.1358.