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arXiv:2508.15892 · Physical Review Research

Breaking global symmetries with locality-preserving operations

Michele Mazzoni, Luca Capizzi, Lorenzo Piroli

Published Dec 17, 2025• 4 citations
arXivHTMLPDF

Abstract

In the general framework of quantum resource theories, one typically only distinguishes between operations that can or cannot generate the resource of interest. In many-body settings, one can further characterize quantum operations based on underlying geometrical constraints, and a natural question is to understand the power of resource-generating operations that preserve locality. In this work, we address this question within the resource theory of asymmetry, which has recently found applications in the study of many-body symmetry-breaking and symmetry-restoration phenomena. We consider symmetries corresponding to both Abelian and non-Abelian compact groups with a homogeneous action on the space of N qubits, focusing on the prototypical examples of U ( 1 ) and SU ( 2 ) . We study the so-called G asymmetry Δ S N G , and present two main results. First, we derive a general bound on the asymmetry that can be generated by locality-preserving operations acting on product states. We prove that, in any spatial dimension, Δ S N G ≤ ( 1 / 2 ) Δ S N G , max [ 1 + o ( 1 ) ] , where Δ S N G , max is the maximum value of the G asymmetry in the full many-body Hilbert space. Second, we show that locality-preserving operations can generate maximal asymmetry, Δ S N G ∼ Δ S N G , max , when applied to symmetric states featuring long-range entanglement. Our results provide a unified perspective on recent studies of asymmetry in many-body physics, highlighting a nontrivial interplay between asymmetry, locality, and entanglement.

DOI
10.1103/gcjj-88j6

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In the general framework of quantum resource theories, one typically only distinguishes between operations that can or cannot generate the resource of interest. In many-body settings, one can further characterize quantum operations based on underlying geometrical constraints, and a natural question is to understand the power of resource-generating operations that preserve locality. In this work, we address this question within the resource theory of asymmetry, which has recently found applications in the study of many-body symmetry-breaking and symmetry-restoration phenomena. We consider symmetries corresponding to both Abelian and non-Abelian compact groups with a homogeneous action on the space of N qubits, focusing on the prototypical examples of U ( 1 ) and SU ( 2 ) . We study the so-called G asymmetry Δ S N G , and present two main results. First, we derive a general bound on the asymmetry that can be generated by locality-preserving operations acting on product states. We prove that, in any spatial dimension, Δ S N G ≤ ( 1 / 2 ) Δ S N G , max [ 1 + o ( 1 ) ] , where Δ S N G , max is the maximum value of the G asymmetry in the full many-body Hilbert space. Second, we show that locality-preserving operations can generate maximal asymmetry, Δ S N G ∼ Δ S N G , max , when applied to symmetric states featuring long-range entanglement. Our results provide a unified perspective on recent studies of asymmetry in many-body physics, highlighting a nontrivial interplay between asymmetry, locality, and entanglement.
Entanglement asymmetry provides a quantitative measure of symmetry breaking in many-body quantum states. Focusing on inhomogeneous U(1)U(1)U(1) charges, such as dipole and multipole moments, we show that the typical asymmetry is bounded by a universal fraction of its maximal value. Multipole charges naturally arise in systems with Hilbert-space fragmentation, where the dynamics splits into exponentially many disconnected sectors. Using the commutant algebra formalism, we generalize entanglement asymmetry to account for fragmentation. While the asymmetry grows logarithmically for conventional symmetries, it can scale extensively in fragmented systems and distinguish classical from quantum fragmentation. We derive general upper bounds for the asymmetry and identify states that saturate them. To study the typical behavior of the asymmetry, we consider the ensemble of random matrix product states. By identifying the bond dimension with an effective time parameter, we qualitatively reproduce recent results on asymmetry dynamics in random quantum circuits, suggesting a universal behavior for the asymmetry of U(1)U(1)U(1) charges in local ergodic systems.
Resource theories provide a universal framework for quantifying properties of quantum states, such as entanglement, magic, athermality, and asymmetry, as resources under various constraints encountered in quantum science. Since realistic settings typically impose several constraints at once, it is important to study multi-resource theories that combine them. Among these, combinations with asymmetry are especially natural, as symmetry is fundamental and ubiquitous in physics. Here we develop a general theory that treats combinations of finite-group asymmetry with a broad range of resource theories in a unified manner. Our key observation is that the effect of combining the two constraints is governed by the accessibility of symmetry information under the underlying resource theory: whether the information needed for the conversion can be extracted from the input and used to control free operations. If this can be done with asymptotically vanishing error, we show that the conversion rate of the combined theory equals the smaller of the rates of the two constituent theories, so combining them incurs no additional cost. This condition holds for LOCC on arbitrary mixed multipartite states, magic-state conversion of multiqubit systems under Clifford symmetries, and Gibbs-preserving operations. Thermal operations, being time-translation covariant, can access only limited symmetry information, giving an additional compatibility condition between the two symmetries. We also analyze symmetry imposed on elementary operations rather than only on the resulting channel. We establish analogous results for LOCC and thermal operations under suitable assumptions, and for magic when the symmetry is represented by Pauli operators. Finally, we show that finite-group symmetry does not reduce the asymptotic ergotropy per copy and hence introduces no new completely passive states.
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.
Symmetry principles have played a crucial role in the development of modern physics and underpin our most fundamental theories of nature. The present thesis is concerned with the analysis of symmetries in the context of quantum information theory. Whenever a quantum mechanical system interacts with its environment it is subjected to decoherence, a process that is typically irreversible and most generally treated abstractly using the concept of quantum operations. If there is an underlying symmetry principle, additional structures emerge. The central question we address is, What are the consequences of global or local gauge symmetry on the structure of many-body quantum processes?This leads us to a diagrammatic framework of decomposing quantum operations into terms that respond to the symmetry principle in particular ways and respect the causal structures involved. We present two core applications. First, we address the interplay between irreversibility and repeatable use of coherent resources under symmetry constraints. Second, we give an information-theoretic perspective on gauging globally symmetric dynamics to a local symmetry applicable even in the presence of irreversibility and thus it goes beyond the usual Lagrangian formulation. Finally, we analyse the departure from conservation laws under symmetric dynamics subject to decoherence.
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.
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)U(1)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.