arXiv:2404.10895 · Journal of Physics A Mathematical and Theoretical

A class of Schwarz qubit maps with diagonal unitary and orthogonal symmetries

Dariusz Chruściński, Bihalan Bhattacharya

Published Aug 30, 2024• 5 citations

Abstract

Abstract A class of unital qubit maps displaying diagonal unitary and orthogonal symmetries is analyzed. Such maps have already found a lot applications in quantum information theory. We provide a complete characterization of this class of maps showing intricate relation between positivity, operator Schwarz inequality, and complete positivity. Finally, it is shown how to generalize the entire picture beyond unital case (so called generalized Schwarz maps). Interestingly, the first example of Schwarz but not completely positive map found by Choi belongs to our class. As a case study we provide a full characterization of Pauli maps. Our analysis leads to generalization of seminal Fujiwara–Algoet conditions for Pauli quantum channels.

DOI: 10.1088/1751-8121/ad75d6

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arXiv:2404.10895 · Journal of Physics A Mathematical and Theoretical

A class of Schwarz qubit maps with diagonal unitary and orthogonal symmetries

Dariusz Chruściński, Bihalan Bhattacharya

Published Aug 30, 2024• 5 citations

arXiv HTML PDF

Abstract

DOI: 10.1088/1751-8121/ad75d6

Related papers

Schwarz maps with symmetry

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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.

Sufficient conditions for the Kadison--Schwarz property of unital positive maps on

M_3

M_3

within the Bloch--Gell--Mann representation and derive explicit analytic sufficient conditions ensuring the Kadison--Schwarz property. The approach exploits unitary equivalence together with structural properties of the Lie algebra

\mathfrak{su}(3)

and does not rely on numerical optimization or semidefinite-programming methods. A key mechanism is the cancellation of contributions associated with antisymmetric structure constants, which reduces the problem to estimates governed solely by the symmetric tensor

d_{ijk}

. The results clarify how the Kadison--Schwarz property can hold under assumptions substantially weaker than complete positivity and yield a structural criterion for the KS property on

M_3

in terms of Bloch parameters.