Schwarz maps with symmetry

TL;DR

This paper applies symmetry theory to classify and analyze Schwarz maps in quantum information, revealing how symmetry influences the structure of non-completely positive maps and confirming the PPT^2 conjecture in various symmetric cases.

Contribution

It provides a systematic classification of U(n)-equivariant Schwarz and completely positive maps, with explicit algebraic conditions and geometric illustrations, advancing understanding of symmetry in quantum maps.

Findings

Complete classification of U(n)-equivariant Schwarz maps

Explicit algebraic inequalities for positivity regions

Verification of PPT^2 conjecture in symmetric cases

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 $\Phi:\mathcal A \to \mathcal B$ 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 $\Phi(X^\ast X) \geq \Phi(X)^\ast \Phi(X)$ 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 $\Phi$ 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.