This thesis studies (generalized) symmetry breaking in states via entanglement measures. Specifically, we study a relative entropy between a given state and a suitably defined symmetric version of the same state; obtained by applying a quantum channel made of the topological operators implementing the symmetry. We study this for both excited states in CFTs as well as symmetry breaking vacua. In the first part, following [1], we study a perturbative version of this asymmetry in CFTs for a U(1) symmetry in various geometries, and study the relaxation of such states in real time while displaying a quantum Mpemba effect. We also study the holography of entanglement asymmetry. In the second part, following [2], we study the symmetrization process for non-invertible symmetries in two spacetime dimensions. We identify the different symmetry algebras that act on a circle or interval hilbert space, and prescribe a universal, unique symmetrization procedure to compute symmetry breaking. We study examples in both CFTs and gapped symmetry breaking states.
This thesis studies (generalized) symmetry breaking in states via entanglement measures. Specifically, we study a relative entropy between a given state and a suitably defined symmetric version of the same state; obtained by applying a quantum channel made of the topological operators implementing the symmetry. We study this for both excited states in CFTs as well as symmetry breaking vacua. In the first part, following [1], we study a perturbative version of this asymmetry in CFTs for a U(1) symmetry in various geometries, and study the relaxation of such states in real time while displaying a quantum Mpemba effect. We also study the holography of entanglement asymmetry. In the second part, following [2], we study the symmetrization process for non-invertible symmetries in two spacetime dimensions. We identify the different symmetry algebras that act on a circle or interval hilbert space, and prescribe a universal, unique symmetrization procedure to compute symmetry breaking. We study examples in both CFTs and gapped symmetry breaking states.
This thesis investigates the use of concepts from quantum information theory to analyze and quantify symmetry breaking in extended quantum systems, particularly focusing on the key measure known as Entanglement Asymmetry. The core of the work introduces and applies Entanglement Asymmetry. This quantity is defined based on the difference in information content between a quantum state and a version of that state where the symmetry has been enforced (or "symmetrized"). It represents an alternative approach to traditional, often problematic, local order parameters for measuring symmetry breaking: asymmetry is zero only if the state perfectly preserves the symmetry. The major contributions of the thesis are structured as follows: Symmetry Breaking in Bulk Critical Systems: The research analyzes Entanglement Asymmetry in two-dimensional Conformal Field Theories (CFTs) where the symmetry is explicitly broken. This work derives the asymptotic behavior of the asymmetry as a function of the subsystem size, extending earlier finite-size results to the critical continuum limit. Boundary Effects and Quenches: The analysis is extended to cases where the symmetry holds in the bulk of a CFT but is broken at the edge by a boundary condition. The thesis provides formulas describing this boundary-induced asymmetry, including power-law corrections whose exponents are linked to the properties of boundary-changing operators. It also explores the time evolution of asymmetry in quantum quench protocols. Generalization of Asymmetry to Modern Symmetries: The concept of Entanglement Asymmetry is generalized to include finite generalized symmetries, such as higher-form and non-invertible symmetries, which are a major topic in modern QFT. This involves establishing a generalized procedure for "symmetrizing" a state without relying on traditional group mathematics. Entanglement in Non-Hermitian Systems: The final chapter shifts focus to explore other entanglement properties, specifically symmetry-resolved entanglement and the entanglement Hamiltonian, within non-Hermitian models and their corresponding non-unitary CFTs.
Entanglement asymmetry is an observable in quantum systems, constructed using quantum-information methods, suited to detecting symmetry breaking in states -- possibly out of equilibrium -- relative to a subsystem. In this paper we define the asymmetry for generalized finite symmetries, including higher-form and noninvertible ones. To this end, we introduce a "symmetrizer" of (reduced) density matrices with respect to the C∗-algebra of symmetry operators acting on the subsystem Hilbert space. We study in detail applications to (1+1)-dimensional quantum field theories: First, we analyze spontaneous symmetry breaking of noninvertible symmetries, confirming that distinct vacua can exhibit different physical properties. Second, we compute the asymmetry of certain excited states in conformal field theories (including the Ising CFT), when the subsystem is either the full circle or an interval therein. The relevant symmetry algebras to consider are the fusion, tube, and strip algebras. Finally, we comment on the case that the symmetry algebra is a (weak) Hopf algebra.
Abstract Entanglement asymmetry is a measure of symmetry breaking in quantum subsystems, inspired by quantum information theory, particularly suited to study out-of-equilibrium states. We study the entanglement asymmetry of a class of excited “coherent states” in conformal quantum field theories with a U(1) symmetry, employing Euclidean path-integral methods with topological symmetry defects and the replica formalism. We compute, at leading order in perturbation theory, the asymmetry for a variety of subsystems, including finite spherical subregions in flat space, in finite volume, and at positive temperature. We also study its Lorentzian time evolution, showcasing the dynamical restoration of the symmetry due to thermalization, as well as the presence of a quantum Mpemba effect. Our results are universal, and apply in any number of dimensions. We also show that the perturbative entanglement asymmetry is related to the Fisher information metric, which has a known holographic dual called the Hollands–Wald canonical energy, and that it is captured by the anti-de-Sitter bulk charge contained in the entanglement wedge.
Entanglement asymmetry is a relative entropy that faithfully diagnoses symmetry breaking in quantum states, possibly within a spatial subregion. In this work, we extend such framework to higher-form symmetries and compute entanglement asymmetry in theories with spontaneously-broken continuous zero- and higher-form symmetries. One of our central results is an entropic Coleman-Mermin-Wagner theorem, for 0- and p-form symmetries, valid also on subregions, which forbids spontaneous breaking of continuous p-form symmetries in spacetime dimensions d≤p+2. Our theorem not only qualifies symmetry breaking, it also quantifies it: spontaneous breaking triggers a nonvanishing entanglement asymmetry that grows monotonically towards the infrared, and counts the number of Goldstone fields. Along the way, we derive standalone results concerning the entanglement entropy and asymmetry of Goldstone bosons and gauge fields. In particular, we find a closed-form expression for the Rényi asymmetries of a compact scalar field on spherical subregions in three and four spacetime dimensions, and for higher-form gauge fields in higher dimensions.
A bstract A state in a quantum system with a given global symmetry, G , can be sensitive to the presence of boundaries, which may either preserve or break this symmetry. In this work, we investigate how conformal invariant boundary conditions influence the G -symmetry breaking through the lens of the entanglement asymmetry, a quantifier of the “distance” between a symmetry-broken state and its symmetrized counterpart. By leveraging 2D boundary conformal field theory (BCFT), we investigate the symmetry breaking for both finite and compact Lie groups. Beyond the leading order term, we also compute the subleading corrections in the subsystem size, highlighting their dependence on the symmetry group G and the BCFT operator content. We further explore the entanglement asymmetry following a global quantum quench, where a symmetry-broken state evolves under a symmetry-restoring Hamiltonian. In this dynamical setting, we compute the entanglement asymmetry by extending the method of images to a BCFT with non-local objects such as invertible symmetry defects.
Quantifying how much a quantum state breaks a symmetry is essential for characterizing phases, nonequilibrium dynamics, and open-system behavior. For mixed states, however, conventional diagnostics of weak symmetry breaking can miss a stronger form of symmetry breaking associated with the possibility of exchanging conserved charges with an environment. We develop a resource theory of strong symmetry breaking by identifying the appropriate free states and free operations, and, for channels on a fixed system, by showing that the latter are precisely the operations realizable without exchanging the conserved charge with the environment. We systematically construct measures of strong symmetry breaking, including strong entanglement asymmetry and covariance matrices of symmetry generators, for a broad class of symmetry groups. We further completely characterize i.i.d. convertibility between arbitrary states under any compact Lie group strong symmetry. This identifies the combinations of resource-theoretically valid measures that quantify strong symmetry breaking for arbitrary states, including all mixed states. In particular, for U(1) symmetry, if the states have positive variances of the conserved quantity and equal strong-symmetry periods, the conversion rate is completely determined by the variance ratio whenever the input is pure or the output is weak symmetric. Thus, for these state conversions, the variance of the conserved quantity plays an operational role analogous to that of entanglement entropy in entanglement theory or quantum Fisher information in the resource theory of weak asymmetry. We further show how weak symmetry breaking is irreversibly converted into strong symmetry breaking in open-system dynamics. We illustrate the framework with examples from quantum field theory, strong-to-weak spontaneous symmetry breaking, and a strong-symmetry analogue of Mpemba-type dynamics.