Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Moscow University Physics Bulletin

On One Conjugate Object in the Symmetric Tensor $$\boldsymbol{C}^{\boldsymbol{*}}$$-Category and the Statistics of Superselection Sectors

Аirat Sitdikov, А. С. Никитин

Published Aug 1, 2023• 1 citation
Publisher

Abstract

The paper presents a model of symmetric tensor $$C^{*}$$ -category with conjugation for an object of dimension $$d=3$$ . It is proved that the constructed conjugate object of the category satisfies the conjugation equations, and different classes of morphisms between the objects in the modelled category are considered and studied. This category allows for the generalization of the $$C^{*}$$ -algebraic model of observables in the presence of superselection rules generated by non-Abelian conjugate charges. As an application of the model, the area of quantum information transfer is explored, where the constraints imposed by the superselection rules need to be taken into account.

DOI
10.3103/s0027134923040173

Related papers

On one conjugate object in the symmetric tensor C*-category and the statistics of superselection sectors

Аirat Sitdikov, А. С. Никитин

In this work, a model of a symmetric tensor 𝐶*-category with conjugation is constructed for the dimension of the object 𝑑 = 3. It is proved that the constructed conjugate object of the category satisfies the conjugation equations. The classes of morphisms between objects of the modeled category are considered and studied, and the algebra of observables in the presence of such an conjugate object is introduced. As an application of the model, we consider the theory of quantum information transfer, where it is necessary to take into account the constraints associated with the superselection rules.

Vestnik Moskovskogo Universiteta Seriya 3 Fizika Astronomiya•Sep 20, 2023•1 citation

Algebraic Model of Non-Abelian Superselection Rules Considering Conjugate Endomorphism

A. S. Nikitin, Аirat Sitdikov

In this paper, we consider an extension of the previously proposed algebraic model and study the constraints of non-Abelian superselection rules on the transfer quantum information, taking into account conjugate endomorphism. The procedure of averaging (over the group $$G=SU(3)$$ ) projectors to the basic states of coherent orthogonal subspaces into which the space of two three-level systems decomposes is considered. Main attention is paid to the superselection structure of the algebra of observables $${}^{0}O_{G}$$ defined by the Cuntz algebra $${}^{0}O_{d=3}$$ (field algebra) containing $${}^{0}O_{G}$$ as a pointwise fixed subalgebra with respect to the action of the gauge group $$G$$ . As an application of the model, we consider the encoding of information using a three-level system and show that information can be transmitted only by those states whose projectors belong to the algebra of observables. These projectors commute with the elements of the representation of the group $$G$$ , and therefore, allow the recipient to restore the obtained information.

Lobachevskii Journal of Mathematics•Jan 1, 2024•1 citation

Superselection rules, bosonization duality in 1 + 1 dimensions, and momentum-space entanglement

Matheus H. Martins Costa, Flavio S. Nogueira, Jeroen van den Brink

We investigate the effects of the presence of conserved charges on the momentum-space entanglement of quantum field theories. We show that if a given model has superselection sectors, then it allows for different notions of momentum modes, each associated with a complete set of commuting observables. Applying this idea to investigate the entanglement of momentum degrees of freedom on both sides of the Abelian bosonization duality, we find that different tensor product partitions are mapped into each other and give explicit examples to sustain our findings. The conditions on which our conclusions may be generalized to other duality transformations, which require the introduction of a notion stricter than the most general possible, are laid and directions for further work are given. Published by the American Physical Society 2024

arXiv:2410.01908•Physical review. D/Physical review. D.•Nov 25, 2024•0 citations

Quantum protocols at presence of nonabelian superselection rules in the framework of algebraic model

Аirat Sitdikov, А. С. Никитин

In this paper, we study the influence of nonabelian superselection rules on the transfer of quantum information with the help of qubits on the base of an algebraic model and formulate quantum protocols. We pay the main attention to the superselection structure of the algebra of observables [Formula: see text] defined by the Cuntz algebra [Formula: see text] (a field algebra) that contains [Formula: see text] as a pointwise fixed subalgebra with respect to the action of the gauge group [Formula: see text]. We prove that it is possible to code information only with the help of states such that projectors on them belong to the algebra of observables. These projectors commute with the elements of the representation of the group [Formula: see text], and therefore allow the recipient to restore the obtained information.

arXiv:2104.05238•International Journal of Quantum Information•Dec 29, 2021•3 citations

Quantum current and holographic categorical symmetry

Tian Lan, Jing-Ren Zhou

We establish the formulation for quantum current. Given a symmetry group G, let \mathcal{C} 𝒞 :=Rep G be its representation category. Physically, symmetry charges are objects of \mathcal{C} 𝒞 and symmetric operators are morphisms in \mathcal{C} 𝒞 . The addition of charges is given by the tensor product of representations. For any symmetric operator O crossing two subsystems, the exact symmetry charge transported by O can be extracted. The quantum current is defined as symmetric operators that can transport symmetry charges over an arbitrary long distance. A quantum current exactly corresponds to an object in the Drinfeld center Z_1(\mathcal{C}) Z 1 ( 𝒞 ) . The condition for quantum currents to be is also specified, which corresponds to condensation of anyons in one higher dimension. To express the local conservation, the internal hom must be used to compute the charge difference, and the framework of enriched category is inevitable. To illustrate these ideas, we develop a rigorous scheme of renormalization in one-dimensional lattice systems and analyse the fixed-point models. It is proved that in the fixed-point models, superconducting quantum currents form a Lagrangian algebra in Z_1(\mathcal{C}) Z 1 ( 𝒞 ) and the boundary-bulk correspondence is verified in the enriched setting. Overall, the quantum current provides a natural physical interpretation to the holographic categorical symmetry.

arXiv:2305.12917•SciPost Physics•Feb 23, 2024•11 citations

Kosaki-Longo index and classification of charges in 2D quantum spin models

Pieter Naaijkens

We consider charge superselection sectors of two-dimensional quantum spin models corresponding to cone localisable charges, and prove that the number of equivalence classes of such charges is bounded by the Kosaki-Longo index of an inclusion of certain observable algebras. To demonstrate the power of this result we apply the theory to the toric code on a 2D infinite lattice. For this model we can compute the index of this inclusion, and conclude that there are four distinct irreducible charges in this model, in accordance with the analysis of the toric code model on compact surfaces. We also give a sufficient criterion for the non-degeneracy of the charge sectors, in the sense that Verlinde's matrix S is invertible.

arXiv:1303.4420•Journal of Mathematical Physics•Aug 1, 2013•16 citations
On One Conjugate Object in the Symmetric Tensor C∗\boldsymbol{C}^{\boldsymbol{*}}C∗-Category and the Statistics of Superselection Sectors
The paper presents a model of symmetric tensor C∗C^{*}C∗ -category with conjugation for an object of dimension d=3d=3d=3 . It is proved that the constructed conjugate object of the category satisfies the conjugation equations, and different classes of morphisms between the objects in the modelled category are considered and studied. This category allows for the generalization of the C∗C^{*}C∗ -algebraic model of observables in the presence of superselection rules generated by non-Abelian conjugate charges. As an application of the model, the area of quantum information transfer is explored, where the constraints imposed by the superselection rules need to be taken into account.
In this work, a model of a symmetric tensor 𝐶*-category with conjugation is constructed for the dimension of the object 𝑑 = 3. It is proved that the constructed conjugate object of the category satisfies the conjugation equations. The classes of morphisms between objects of the modeled category are considered and studied, and the algebra of observables in the presence of such an conjugate object is introduced. As an application of the model, we consider the theory of quantum information transfer, where it is necessary to take into account the constraints associated with the superselection rules.
In this paper, we consider an extension of the previously proposed algebraic model and study the constraints of non-Abelian superselection rules on the transfer quantum information, taking into account conjugate endomorphism. The procedure of averaging (over the group G=SU(3)G=SU(3)G=SU(3) ) projectors to the basic states of coherent orthogonal subspaces into which the space of two three-level systems decomposes is considered. Main attention is paid to the superselection structure of the algebra of observables 0OG{}^{0}O_{G}0OG​ defined by the Cuntz algebra 0Od=3{}^{0}O_{d=3}0Od=3​ (field algebra) containing 0OG{}^{0}O_{G}0OG​ as a pointwise fixed subalgebra with respect to the action of the gauge group GGG . As an application of the model, we consider the encoding of information using a three-level system and show that information can be transmitted only by those states whose projectors belong to the algebra of observables. These projectors commute with the elements of the representation of the group GGG , and therefore, allow the recipient to restore the obtained information.
We investigate the effects of the presence of conserved charges on the momentum-space entanglement of quantum field theories. We show that if a given model has superselection sectors, then it allows for different notions of momentum modes, each associated with a complete set of commuting observables. Applying this idea to investigate the entanglement of momentum degrees of freedom on both sides of the Abelian bosonization duality, we find that different tensor product partitions are mapped into each other and give explicit examples to sustain our findings. The conditions on which our conclusions may be generalized to other duality transformations, which require the introduction of a notion stricter than the most general possible, are laid and directions for further work are given. Published by the American Physical Society 2024
In this paper, we study the influence of nonabelian superselection rules on the transfer of quantum information with the help of qubits on the base of an algebraic model and formulate quantum protocols. We pay the main attention to the superselection structure of the algebra of observables [Formula: see text] defined by the Cuntz algebra [Formula: see text] (a field algebra) that contains [Formula: see text] as a pointwise fixed subalgebra with respect to the action of the gauge group [Formula: see text]. We prove that it is possible to code information only with the help of states such that projectors on them belong to the algebra of observables. These projectors commute with the elements of the representation of the group [Formula: see text], and therefore allow the recipient to restore the obtained information.
We establish the formulation for quantum current. Given a symmetry group G, let \mathcal{C} 𝒞 :=Rep G be its representation category. Physically, symmetry charges are objects of \mathcal{C} 𝒞 and symmetric operators are morphisms in \mathcal{C} 𝒞 . The addition of charges is given by the tensor product of representations. For any symmetric operator O crossing two subsystems, the exact symmetry charge transported by O can be extracted. The quantum current is defined as symmetric operators that can transport symmetry charges over an arbitrary long distance. A quantum current exactly corresponds to an object in the Drinfeld center Z_1(\mathcal{C}) Z 1 ( 𝒞 ) . The condition for quantum currents to be is also specified, which corresponds to condensation of anyons in one higher dimension. To express the local conservation, the internal hom must be used to compute the charge difference, and the framework of enriched category is inevitable. To illustrate these ideas, we develop a rigorous scheme of renormalization in one-dimensional lattice systems and analyse the fixed-point models. It is proved that in the fixed-point models, superconducting quantum currents form a Lagrangian algebra in Z_1(\mathcal{C}) Z 1 ( 𝒞 ) and the boundary-bulk correspondence is verified in the enriched setting. Overall, the quantum current provides a natural physical interpretation to the holographic categorical symmetry.
We consider charge superselection sectors of two-dimensional quantum spin models corresponding to cone localisable charges, and prove that the number of equivalence classes of such charges is bounded by the Kosaki-Longo index of an inclusion of certain observable algebras. To demonstrate the power of this result we apply the theory to the toric code on a 2D infinite lattice. For this model we can compute the index of this inclusion, and conclude that there are four distinct irreducible charges in this model, in accordance with the analysis of the toric code model on compact surfaces. We also give a sufficient criterion for the non-degeneracy of the charge sectors, in the sense that Verlinde's matrix S is invertible.