© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:1905.10487 · Journal of High Energy Physics

Entanglement entropy and superselection sectors. Part I. Global symmetries

Horacio Casini, Marina Huerta, Javier M. Magán, Diego Pontello

Published Feb 1, 2020• 87 citations

Abstract

Abstract Some quantum field theories show, in a fundamental or an effective manner, an alternative between a loss of duality for algebras of operators corresponding to complementary regions, or a loss of additivity. In this latter case, the algebra contains some operator that is not generated locally, in the former, the entropies of complementary regions do not coincide. Typically, these features are related to the incompleteness of the operator content of the theory, or, in other words, to the existence of superselection sectors. We review some aspects of the mathematical literature on superselection sectors aiming attention to the physical picture and focusing on the consequences for entanglement entropy (EE). For purposes of clarity, the whole discussion is divided into two parts according to the superselection sectors classification: the present part I is devoted to superselection sectors arising from global symmetries, and the forthcoming part II will consider those arising from local symmetries. Under this perspective, here restricted to global symmetries, we study in detail different cases such as models with finite and Lie group symmetry as well as with spontaneous symmetry breaking or excited states. We illustrate the general results with simple examples. As an important application, we argue the features of holographic entanglement entropy correspond to a picture of an sub-theory with a large number of superselection sectors and suggest some ways in which this identification could be made more precise.

DOI: 10.1007/jhep02(2020)014

Related papers

Entanglement entropy and duality

Đorđe Radičević

Using the algebraic approach to entanglement entropy, we study several dual pairs of lattice theories and show how the entropy is completely preserved across each duality. Our main result is that a maximal algebra of observables in a region typically dualizes to a non-maximal algebra in a dual region. In particular, we show how the usual notion of tracing out external degrees of freedom dualizes to a tracing out coupled to an additional summation over superselection sectors. We briefly comment on possible extensions of our results to more intricate dualities, including holographic ones.

Journal of High Energy Physics•Nov 1, 2016•29 citations

Aspects of Entanglement Entropy in Algebraic Quantum Field Theory

Diego Pontello

In this thesis, we study aspects of entanglement theory of quantum field theories from an algebraic point of view. The main motivation is to gain insights about the general structure of the entanglement in QFT, towards a definition of an entropic version of QFT. In the opposite direction, we are also interested in exploring any consequence of the entanglement in algebraic QFT. This may help us to reveal unknown features of QFT, with the final aim of finding a dynamical principle which allows us to construct non-trivial and rigorous models of QFT. The algebraic approach is the natural framework to define and study entanglement in QFT, and hence, to pose the above inquiries. After a self-contained review of algebraic QFT and quantum information theory in operator algebras, we focus on our results. We compute, in a mathematically rigorous way, exact solutions of entanglement measures and modular Hamiltonians for specific QFT models, using algebraic tools from modular theory of von Neumann algebras. These calculations show explicitly non-local features of modular Hamiltonians and help us to solve ambiguities that arise in other non-rigorous computations. We also study aspects of entanglement entropy in theories having superselection sectors coming from global symmetries. We follow the algebraic perspective of Doplicher, Haag, and Roberts. In this way, we find an entropic order parameter that "measures" the size of the symmetry group, which is made out of a difference of two mutual informations. Moreover, we identify the main operators that take account of such a difference, and we obtain a new quantum information quantity, the entropic certainty relation, involving algebras containing such operators. This certainty relation keeps an intrinsic connection with subfactor theory of von Neumann algebras.

arXiv:2005.13975•May 28, 2020•3 citations

Entanglement Entropy and Duality

Djordje Radicevic

Using the algebraic approach to entanglement entropy, we study several dual pairs of lattice theories and show how the entropy is completely preserved across each duality. Our main result is that a maximal algebra of observables in a region typically dualizes to a non-maximal algebra in a dual region. In particular, we show how the usual notion of tracing out external degrees of freedom dualizes to a tracing out coupled to an additional summation over superselection sectors. We briefly comment on possible extensions of our results to more intricate dualities, including holographic ones.

arXiv:1605.09396•May 30, 2016•6 citations

Generalized entanglement entropy

Marika Taylor

We discuss two measures of entanglement in quantum field theory and their holographic realizations. For field theories admitting a global symmetry, we introduce a global symmetry entanglement entropy, associated with the partitioning of the symmetry group. This quantity is proposed to be related to the generalized holographic entanglement entropy defined via the partitioning of the internal space of the bulk geometry. Thesecond measure of quantum field theory entanglement is the field space entanglement entropy, obtained by integrating out a subset of the quantum fields. We argue that field space entanglement entropy cannot be precisely realised geometrically in a holographic dual. However, for holographic geometries with interior decoupling regions, the differential entropy provides a close analogue to the field space entanglement entropy. We derive generic descriptions of such inner throat regions in terms of gravity coupled to massive scalars and show how the differential entropy in the throat captures features of the field space entanglement entropy.

arXiv:1507.06410•Journal of High Energy Physics•Jul 1, 2016•25 citations

Instability of universal terms in the entanglement entropy

Marina Huerta, Guido van der Velde

The role of symmetries in what concerns entanglement entropy has been extensively explored in the last years and revealed a profound connection with the quantum field theory's algebraic structure. Recently, it was found that some universal contributions to the entanglement entropy and mutual information may be nonuniquely defined in theories with generalized symmetries. Here, we study this issue in detail in the particular case of the entanglement entropy of the Maxwell theory in ($2+1$) dimensions for rotationally symmetric regions. In this setup, the problem can be dimensionally reduced to a half-line. We find that the only difference between the reduced problem for the Maxwell field and the reduced scalar free field stems from the Fourier angular $n=0$ mode. This simplification allows us to check explicitly the many issues that characterize models with broken global symmetries. Namely, we manifestly show that the additive algebras break Haag duality, and single out the nonlocal operators that are responsible for the failure of this property. More interestingly, we present concrete lattice realizations that confirm that the logarithmic ``universal'' term of the Maxwell entanglement entropy for disks depends on the details of the algebra assignation. This ambiguity hinders the identification of possible topological contributions characteristic of models with generalized symmetries and tarnishes its universal character. We further calculate the Maxwell mutual information for two nearly complementary concentric disks. We obtain the expected universal contribution with a log-log dependence and check that, unlike entropy, this is stable. Accordingly, this supports mutual information as the appropriate probe to sense additivity-duality breaking and the consequent universal topological contributions.

arXiv:2204.09464•Physical review. D/Physical review. D.•Jun 30, 2022•11 citations

Entanglement entropy and superselection sectors. Part I. Global symmetries

Horacio Casini, Marina Huerta, Javier M. Magán +1 more

•Feb 1, 2020•16 citations

Abstract Some quantum field theories show, in a fundamental or an effective manner, an alternative between a loss of duality for algebras of operators corresponding to complementary regions, or a loss of additivity. In this latter case, the algebra contains some operator that is not generated locally, in the former, the entropies of complementary regions do not coincide. Typically, these features are related to the incompleteness of the operator content of the theory, or, in other words, to the existence of superselection sectors. We review some aspects of the mathematical literature on superselection sectors aiming attention to the physical picture and focusing on the consequences for entanglement entropy (EE). For purposes of clarity, the whole discussion is divided into two parts according to the superselection sectors classification: the present part I is devoted to superselection sectors arising from global symmetries, and the forthcoming part II will consider those arising from local symmetries. Under this perspective, here restricted to global symmetries, we study in detail different cases such as models with finite and Lie group symmetry as well as with spontaneous symmetry breaking or excited states. We illustrate the general results with simple examples. As an important application, we argue the features of holographic entanglement entropy correspond to a picture of an sub-theory with a large number of superselection sectors and suggest some ways in which this identification could be made more precise.

Using the algebraic approach to entanglement entropy, we study several dual pairs of lattice theories and show how the entropy is completely preserved across each duality. Our main result is that a maximal algebra of observables in a region typically dualizes to a non-maximal algebra in a dual region. In particular, we show how the usual notion of tracing out external degrees of freedom dualizes to a tracing out coupled to an additional summation over superselection sectors. We briefly comment on possible extensions of our results to more intricate dualities, including holographic ones.

In this thesis, we study aspects of entanglement theory of quantum field theories from an algebraic point of view. The main motivation is to gain insights about the general structure of the entanglement in QFT, towards a definition of an entropic version of QFT. In the opposite direction, we are also interested in exploring any consequence of the entanglement in algebraic QFT. This may help us to reveal unknown features of QFT, with the final aim of finding a dynamical principle which allows us to construct non-trivial and rigorous models of QFT. The algebraic approach is the natural framework to define and study entanglement in QFT, and hence, to pose the above inquiries. After a self-contained review of algebraic QFT and quantum information theory in operator algebras, we focus on our results. We compute, in a mathematically rigorous way, exact solutions of entanglement measures and modular Hamiltonians for specific QFT models, using algebraic tools from modular theory of von Neumann algebras. These calculations show explicitly non-local features of modular Hamiltonians and help us to solve ambiguities that arise in other non-rigorous computations. We also study aspects of entanglement entropy in theories having superselection sectors coming from global symmetries. We follow the algebraic perspective of Doplicher, Haag, and Roberts. In this way, we find an entropic order parameter that "measures" the size of the symmetry group, which is made out of a difference of two mutual informations. Moreover, we identify the main operators that take account of such a difference, and we obtain a new quantum information quantity, the entropic certainty relation, involving algebras containing such operators. This certainty relation keeps an intrinsic connection with subfactor theory of von Neumann algebras.

Using the algebraic approach to entanglement entropy, we study several dual pairs of lattice theories and show how the entropy is completely preserved across each duality. Our main result is that a maximal algebra of observables in a region typically dualizes to a non-maximal algebra in a dual region. In particular, we show how the usual notion of tracing out external degrees of freedom dualizes to a tracing out coupled to an additional summation over superselection sectors. We briefly comment on possible extensions of our results to more intricate dualities, including holographic ones.

We discuss two measures of entanglement in quantum field theory and their holographic realizations. For field theories admitting a global symmetry, we introduce a global symmetry entanglement entropy, associated with the partitioning of the symmetry group. This quantity is proposed to be related to the generalized holographic entanglement entropy defined via the partitioning of the internal space of the bulk geometry. Thesecond measure of quantum field theory entanglement is the field space entanglement entropy, obtained by integrating out a subset of the quantum fields. We argue that field space entanglement entropy cannot be precisely realised geometrically in a holographic dual. However, for holographic geometries with interior decoupling regions, the differential entropy provides a close analogue to the field space entanglement entropy. We derive generic descriptions of such inner throat regions in terms of gravity coupled to massive scalars and show how the differential entropy in the throat captures features of the field space entanglement entropy.

The role of symmetries in what concerns entanglement entropy has been extensively explored in the last years and revealed a profound connection with the quantum field theory's algebraic structure. Recently, it was found that some universal contributions to the entanglement entropy and mutual information may be nonuniquely defined in theories with generalized symmetries. Here, we study this issue in detail in the particular case of the entanglement entropy of the Maxwell theory in (

2+1

) dimensions for rotationally symmetric regions. In this setup, the problem can be dimensionally reduced to a half-line. We find that the only difference between the reduced problem for the Maxwell field and the reduced scalar free field stems from the Fourier angular

n=0

mode. This simplification allows us to check explicitly the many issues that characterize models with broken global symmetries. Namely, we manifestly show that the additive algebras break Haag duality, and single out the nonlocal operators that are responsible for the failure of this property. More interestingly, we present concrete lattice realizations that confirm that the logarithmic ``universal'' term of the Maxwell entanglement entropy for disks depends on the details of the algebra assignation. This ambiguity hinders the identification of possible topological contributions characteristic of models with generalized symmetries and tarnishes its universal character. We further calculate the Maxwell mutual information for two nearly complementary concentric disks. We obtain the expected universal contribution with a log-log dependence and check that, unlike entropy, this is stable. Accordingly, this supports mutual information as the appropriate probe to sense additivity-duality breaking and the consequent universal topological contributions.