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.
In recent years, the traditional notion of symmetry in quantum theory was expanded to so-called generalised or categorical symmetries, which, unlike ordinary group symmetries, may be non-invertible. This appears to be at odds with Wigner's theorem, which requires quantum symmetries to be implemented by (anti)unitary -- and hence invertible -- operators in order to preserve probabilities. We resolve this puzzle for (higher) fusion category symmetries
C by proposing that, instead of acting by unitary operators on a fixed Hilbert space, symmetry defects in
C act as isometries between distinct Hilbert spaces constructed from twisted sectors. As a result, we find that non-invertible symmetries naturally act as trace-preserving quantum channels. Crucially, our construction relies on the symmetry category
C being unitary. We illustrate our proposal through several examples that include Tambara-Yamagami, Fibonacci, and Yang-Lee as well as higher categorical symmetries.
Generalized symmetries often appear in the form of emergent symmetries in low energy effective descriptions of quantum many-body systems. Non-invertible symmetries are a particularly exotic class of generalized symmetries, in that they are implemented by transformations that do not form a group. Such symmetries appear in large families of gapless states of quantum matter and constrain their low-energy dynamics. To provide a UV-complete description of such symmetries, it is useful to construct lattice models that respect these symmetries exactly. In this paper, we discuss two families of one-dimensional lattice Hamiltonians with finite on-site Hilbert spaces: one with (invertible) S_3 S 3 symmetry and the other with non-invertible \mathsf{Rep}(S_3) 𝖱 𝖾 𝗉 ( S 3 ) symmetry. Our models are largely analytically tractable and demonstrate all possible spontaneous symmetry breaking patterns of these symmetries. Moreover, we use numerical techniques to study the nature of continuous phase transitions between the different symmetry-breaking gapped phases associated with both symmetries. Both models have self-dual lines, where the models are enriched by so-called intrinsically non-invertible symmetries generated by Kramers-Wannier-like duality transformations. We provide explicit lattice operators that generate these non-invertible self-duality symmetries. We show that the enhanced symmetry at the self-dual lines is described by a 2+1d symmetry-topological-order (SymTO) of type JK_4\boxtimes \overline{JK}_4 J K 4 ⊠ J K ¯ 4 . The condensable algebras of the SymTO determine the allowed gapped and gapless states of the self-dual S_3 S 3 -symmetric and \mathsf{Rep}(S_3) 𝖱 𝖾 𝗉 ( S 3 ) -symmetric models.
Modulated symmetries are internal symmetries that act in a non-uniform, spatially modulated way and are generalizations of, for example, dipole symmetries. In this paper, we systematically study the gauging of finite Abelian modulated symmetries in {1+1} 1 + 1 dimensions. Working with local Hamiltonians of spin chains, we explore the dual symmetries after gauging and their potential new spatial modulations. We establish sufficient conditions for the existence of an isomorphism between the modulated symmetries and their dual, naturally implemented by lattice reflections. For instance, in systems of prime qudits, translation invariance guarantees this isomorphism. For non-prime qudits, we show using techniques from ring theory that this isomorphism can also exist, although it is not guaranteed by lattice translation symmetry alone. From this isomorphism, we identify new Kramers-Wannier dualities and construct related non-invertible reflection symmetry operators using sequential quantum circuits. Notably, this non-invertible reflection symmetry exists even when the system lacks ordinary reflection symmetry. Throughout the paper, we illustrate these results using various simple toy models.
The theory of symmetry of quantum mechanical systems is applied to study the structure and properties of several classes of relevant maps in quantum information theory: CPTP, PPT and Schwarz maps. First, we develop the general structure that equivariant maps
Φ:A→B between
C∗-algebras satisfy. Then, we undertake a systematic study of unital, Hermiticity-preserving maps that are equivariant under natural unitary group actions. Schwarz maps satisfy Kadison's inequality
Φ(X∗X)≥Φ(X)∗Φ(X) and form an intermediate class between positive and completely positive maps. We completely classify
U(n)-equivariant on
Mn(C) and determine those that are completely positive and Schwarz. Partial classifications are then obtained for the weaker
DU(n)-equivariance (diagonal unitary symmetry) and for tensor-product symmetries
U(n1)⊗U(n2). In each case, the parameter regions where
Φ is Schwarz or completely positive are described by explicit algebraic inequalities, and their geometry is illustrated. Finally, we further show that the
U(n)-equivariant family satisfies
PPT⟺EB, while the
DU(2), symmetric
DU(3),
U(2)⊗U(2) and
U(2)⊗U(3), families obey the
PPT2 conjecture through a direct symmetry argument. These results reveal how group symmetry controls the structure of non-completely positive maps and provide new concrete examples where the
PPT2 property holds.
A bstract The Kramers-Wannier self-duality of critical quantum chains is examined from the perspective of model wave functions. We demonstrate, using the transverse-field Ising chain and the 3-state Potts chain as examples, that the symmetry operator for the Kramers-Wannier self-duality follows in a simple and direct way from a ‘generalised’ translation symmetry of the model wave function in the anyonic fusion basis. This translation operation, in turn, comprises a sequence of F -moves in the underlying fusion category. The symmetry operator thus obtained naturally admits the form of a matrix product operator and obeys non-invertible fusion rules. The findings reveal an intriguing connection between the (non-invertible) translation symmetry on the lattice and topological aspects of the conformal field theory describing the scaling limit.