Vestnik Moskovskogo Universiteta Seriya 3 Fizika Astronomiya

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

Аirat Sitdikov, А. С. Никитин

Published Sep 20, 2023• 1 citation

Abstract

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.

DOI: 10.55959/msu0579-9392.78.2340101

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Vestnik Moskovskogo Universiteta Seriya 3 Fizika Astronomiya

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

Аirat Sitdikov, А. С. Никитин

Published Sep 20, 2023• 1 citation

Publisher

Abstract

DOI: 10.55959/msu0579-9392.78.2340101

Related papers

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

Аirat Sitdikov, А. С. Никитин

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.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.

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On One Conjugate Object in the Symmetric Tensor

\boldsymbol{C}^{\boldsymbol{*}}

-Category and the Statistics of Superselection Sectors

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.

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.

The article arXiv:2410.21454 constructed a braided

\mathrm{W^*}

-tensor category of superselection sectors associated to a net of von Neumann algebras on a suitably geometric poset using localized and transportable endomorphisms. In this article, we construct an equivalent tensor product and braiding using the Connes fusion tensor product without localizing, using ideas from arXiv:1812.04470 for conformal nets. As an application, we prove that the category of superselection sectors associated to the 2D braided quantum spin system constructed from a unitary braided fusion category

\mathcal{B}

in arXiv:2506.19969 is equivalent to

\mathsf{Hilb}(\mathcal{B})^{\rm op}

, the opposite of the unitary ind-completion of

\mathcal{B}

, as a braided

\mathrm{W^*}

-tensor category.

\mathbb {R}^2

R 2 , we weaken our conditions to a bounded spread version of Haag duality and obtain similar results. We show that intertwined nets of algebras have isomorphic braided tensor categories of superselection sectors. Finally, we show that the categories constructed here are equivalent to those constructed by Naaijkens and Ogata for certain 2D quantum spin systems.