Schwarz maps with symmetry

• URL: http://arxiv.org/abs/2601.02282v1
• Date: Mon, 05 Jan 2026 17:11:03 GMT
• Title: Schwarz maps with symmetry
• Authors: Alfonso García-Velo, Alberto Ibort,
• Abstract summary: 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.<n>We develop the general structure that equivariant maps $:mathcal A to mathcal B$ between $Cast$-algebras satisfy.<n>We then undertake a systematic study of unital, Hermiticity-preserving maps that are equivariant under natural unitary group actions.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $Φ:\mathcal A \to \mathcal B$ between $C^\ast$-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 $Φ(X^\ast X) \geq Φ(X)^\ast Φ(X)$ and form an intermediate class between positive and completely positive maps. We completely classify $U(n)$-equivariant on $M_n(\mathbb C)$ 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 $U(n_1) \otimes U(n_2)$. 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 $\mathrm{PPT} \iff \mathrm{EB}$, while the $DU(2)$, symmetric $DU(3)$, $U(2) \otimes U(2)$ and $U(2) \otimes U(3)$, families obey the $\mathrm{PPT}^2$ 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 $\mathrm{PPT}^2$ property holds.

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