Quantifying how much a quantum state breaks a symmetry is essential for characterizing phases, nonequilibrium dynamics, and open-system behavior. Quantum resource theory provides a rigorous operational framework to define and characterize such quantifiers of symmetry-breaking. As a starter, we exemplify the usefulness of resource theory by noting that second-Rényi entanglement asymmetry can increase under symmetric operations, and hence is not a resource monotone, and should not solely be used to capture Quantum Mpemba effect. More importantly, motivated by mixed-state physics where weak and strong symmetries are inequivalent, we formulate a new resource theory tailored to strong symmetry, identifying free states and strong-covariant operations. This framework systematically identifies quantifiers of strong symmetry breaking for a broad class of symmetry groups, including a strong entanglement asymmetry. A particularly transparent structure emerges for U(1) symmetry, where the resource theory for the strong symmetry breaking has a completely parallel structure to the entanglement theory: the variance of the conserved quantity fully characterizes the asymptotic manipulation of strong symmetry breaking. By connecting this result to the knowledge of the geometry of quantum state space, we obtain a quantitative framework to track how weak symmetry breaking is irreversibly converted into strong symmetry breaking in open quantum systems. We further propose extensions to generalized symmetries and illustrate the qualitative impact of strong symmetry breaking in analytically tractable QFT examples and applications.
Quantifying how much a quantum state breaks a symmetry is essential for characterizing phases, nonequilibrium dynamics, and open-system behavior. Quantum resource theory provides a rigorous operational framework to define and characterize such quantifiers of symmetry-breaking. As a starter, we exemplify the usefulness of resource theory by noting that second-Rényi entanglement asymmetry can increase under symmetric operations, and hence is not a resource monotone, and should not solely be used to capture Quantum Mpemba effect. More importantly, motivated by mixed-state physics where weak and strong symmetries are inequivalent, we formulate a new resource theory tailored to strong symmetry, identifying free states and strong-covariant operations. This framework systematically identifies quantifiers of strong symmetry breaking for a broad class of symmetry groups, including a strong entanglement asymmetry. A particularly transparent structure emerges for U(1) symmetry, where the resource theory for the strong symmetry breaking has a completely parallel structure to the entanglement theory: the variance of the conserved quantity fully characterizes the asymptotic manipulation of strong symmetry breaking. By connecting this result to the knowledge of the geometry of quantum state space, we obtain a quantitative framework to track how weak symmetry breaking is irreversibly converted into strong symmetry breaking in open quantum systems. We further propose extensions to generalized symmetries and illustrate the qualitative impact of strong symmetry breaking in analytically tractable QFT examples and applications.
Dynamical symmetries in closed systems lead to conservation laws and to selection rules that are based on those conservation laws. However, conservation laws are not applica- ble to open systems that undergo irreversible transformations. More general selection rules are needed to determine whether, given two states, the transition from one state to the other is possible. Characterizing the asymmetry properties of quantum states using quantum information theoretical tools and methods is particularly fruitful for this purpose. This thesis is concerned with the problem of finding new general selection rules that hold for both open and closed systems. The usual approach to this problem relies heavily on group theory and involves a detailed study of the structure of the symmetry group. In this thesis, we approach the problem in a completely new way. Our approach is to use entanglement to investigate the asymmetry properties of quantum states. In order to do that, we embed the system’s Hilbert space in a larger tensor product Hilbert space, whereby all symmetric states are mapped to separable states, asymmetric states are mapped to entangled states, and the symmetric transformations between two states are replaced by local operations on their bipartite images. Our method is not restricted to only a specific group but applies, in general, to all symmetries that are associated with semi-simple compact Lie groups and their associated Lie algebras. The mapping of the original states to bipartite states that act on the larger Hilbert space enables us to use the well studied theory of entanglement to investigate the consequences of dynamic symmetries. For example, the monotonicity condition on measures of entanglement provide us with new selection rules. Under reversible transfor- mations, the entanglement of the bipartite image states becomes a conserved quantity. These entanglement-based conserved quantities are new and different from the conserved quantities based on expectation values of the Hamiltonian symmetry generators.
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.
In recent years, in quantum information theory, there has been a remarkable development in the general theoretical framework for studying symmetry in dynamics. This development, called resource theory of asymmetry, is expected to have a wide range of applications, from accurate time measurements to black-hole physics. Despite its importance, the foundation of resource theory of asymmetry still has room for expansion. An important problem is in quantifying the amount of resource. When the symmetry prescribed U(1) or R, i.e., with a single conserved quantity, the quantum Fisher information is known as a resource measure that has suitable properties and a clear physical meaning related to quantum fluctuation of the conserved quantity. However, it is not clear what is the resource measure with such suitable properties when a general symmetry prevails for which there are multiple conserved quantities. The purpose of this paper is to fill this gap. Specifically, we show that the quantum Fisher information matrix is a resource measure whenever a connected linear Lie group describes the symmetry. We also consider the physical meaning of this matrix and see which properties that the quantum Fisher information has when the symmetry is described by U(1) or R can be inherited by the quantum Fisher information matrix.
Symmetry in mixed quantum states can manifest in two distinct forms: , where each individual pure state in the quantum ensemble is symmetric with the same charge, and , which applies only to the entire ensemble. This paper explores a novel type of spontaneous symmetry breaking (SSB) where a strong symmetry is broken to a weak one. While the SSB of a weak symmetry is measured by the long-ranged two-point correlation function, the strong-to-weak SSB (SWSSB) is measured by the . We prove that SWSSB is a universal property of mixed-state quantum phases, in the sense that the phenomenon of SWSSB is robust against symmetric low-depth local quantum channels. We also show that the symmetry breaking is “spontaneous” in the sense that the effect of a local symmetry-breaking measurement cannot be recovered locally. We argue that a thermal state at a nonzero temperature in the canonical ensemble (with fixed symmetry charge) should have spontaneously broken strong symmetry. Additionally, we study nonthermal scenarios where decoherence induces SWSSB, leading to phase transitions described by classical statistical models with bond randomness. In particular, the SWSSB transition of a decohered Ising model can be viewed as the “ungauged” version of the celebrated toric-code decodability transition. We confirm that, in the decohered Ising model, the SWSSB transition defined by the fidelity correlator is the only physical transition in terms of channel recoverability. We also comment on other (inequivalent) definitions of SWSSB, through correlation functions with higher Rényi indices.
Symmetry breaking underlies diverse phenomena from phase transitions in condensed matter to fundamental interactions in gauge theories. Despite many proposed indicators, a general quantification of symmetry breaking that is faithful, computable, and valid in the thermodynamic limit has remained elusive. Here, within quantum resource theory, we propose the quantum Fisher information (QFI) as such a measure. We demonstrate its utility by computing QFI for paradigmatic models: in the BCS superconductor, the QFI counts the number of Cooper pairs; in the transverse-field XY spin chains, it captures topological phase transition that has no local order parameter; and in quantum quench dynamics, it allows us to exactly derive the microscopic origin and conditions of the quantum Mpemba effect in terms of excitation propagation, including in the thermodynamic limit--beyond the reach of previous analyses. Our results show that the QFI, which is a complete resource monotone in the resource theory of asymmetry that plays the role of entanglement entropy in entanglement theory, faithfully captures symmetry breaking in condensed-matter systems. These results highlight the QFI as a universal and physically meaningful diagnostic of symmetry breaking in both equilibrium and non-equilibrium quantum many-body systems.
Quantifying physical concepts in terms of the ultimate performance of a given task has been central to theoretical progress, as illustrated by thermodynamic entropy and entanglement entropy, which quantify irreversibility and quantum correlations, respectively. Symmetry breaking is equally universal yet lacks such an operational quantification. While an operational characterization of symmetry breaking through asymptotic state-conversion efficiency is a central goal of the resource theory of asymmetry (RTA), such a characterization has so far been completed only for the U(1) group among continuous symmetries. Here, we identify the complete measure of symmetry breaking for a general continuous symmetry described by any compact Lie group. Specifically, we show that the asymptotic conversion rate between many copies of pure states in RTA is determined by the quantum geometric tensor, thereby establishing it as the complete measure of symmetry breaking. As an immediate consequence of our conversion rate formula, we also resolve the Marvian-Spekkens conjecture on conditions for reversible conversion in RTA, which has remained unproven for over a decade. Leveraging the connection between symmetry breaking and the theory of quantum reference frames, we also systematically introduce a standardized reference state for frameness based on our asymptotic conversion theory. In addition, by applying our analysis to a standard quantum-thermodynamic scenario, we show that asymptotic state conversion in contact with heat baths generally requires macroscopic coherence in the thermodynamic limit.