Journal of Physics Conference Series

Covariant Schwarz maps in M2(ℂ)

Dariusz Chruściński

Published Dec 1, 2024• 0 citations

Abstract

Abstract We analyze a class of qubit maps displaying diagonal unitary and orthogonal symmetries. For unital maps we characterize all covariant maps satisfying an operator Schwarz inequality. In particular well known Pauli maps are completely characterized. Going beyond the unital case we consider recently proposed generalizations of Schwarz inequality and provide the corresponding necessary and sufficient conditions for the entire class of covariant maps.

DOI: 10.1088/1742-6596/2912/1/012020

Related papers

Schwarz maps with symmetry

Alfonso García-Velo, Alberto Ibort

quant-pharXiv:2601.02282Jan 6, 20260 citations

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

Dariusz Chruściński, Bihalan Bhattacharya

arXiv:2404.10895Journal of Physics A Mathematical and TheoreticalAug 30, 20245 citations

Universal constraint for relaxation rates of semigroups of qubit Schwarz maps

Dariusz Chruściński, Gen Kimura, Farrukh Maksutovich Mukhamedov

arXiv:2401.05051Journal of Physics A Mathematical and TheoreticalApr 9, 20244 citations

Diagonal Unitary Covariant Superchannels

Dariusz Chruściński, Vivek Pandey, Sohail

quant-pharXiv:2512.24389Journal of Mathematical PhysicsDec 31, 20250 citations

Diagonal unitary and orthogonal symmetries in quantum theory

Satvik Singh, Ion Nechita

arXiv:2010.07898QuantumAug 9, 202127 citations

Sufficient conditions for the Kadison--Schwarz property of unital positive maps on $M_3$

Adam Rutkowski

quant-pharXiv:2512.18900Dec 23, 20250 citations

Journal of Physics Conference Series

Covariant Schwarz maps in M2(ℂ)

Dariusz Chruściński

Published Dec 1, 2024• 0 citations

Publisher

Abstract

DOI: 10.1088/1742-6596/2912/1/012020

Related papers

Schwarz maps with symmetry

Alfonso García-Velo, Alberto Ibort

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.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.

quant-pharXiv:2601.02282Jan 6, 20260 citations

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

Dariusz Chruściński, Bihalan Bhattacharya

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.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.

arXiv:2404.10895Journal of Physics A Mathematical and TheoreticalAug 30, 20245 citations

Universal constraint for relaxation rates of semigroups of qubit Schwarz maps

Dariusz Chruściński, Gen Kimura, Farrukh Maksutovich Mukhamedov

Abstract Unital qubit Schwarz maps interpolate between positive and completely positive maps. It is shown that the relaxation rates of the qubit semigroups of unital maps enjoying the Schwarz property satisfy a universal constraint, which provides a modification of the corresponding constraint known for completely positive semigroups. As an illustration, we consider two paradigmatic qubit semigroups: Pauli dynamical maps and phase-covariant dynamics. This result has two interesting implications: it provides a universal constraint for the spectra of qubit Schwarz maps and gives rise to a necessary condition for a Schwarz qubit map to be Markovian.Abstract Unital qubit Schwarz maps interpolate between positive and completely positive maps. It is shown that the relaxation rates of the qubit semigroups of unital maps enjoying the Schwarz property satisfy a universal constraint, which provides a modification of the corresponding constraint known for completely positive semigroups. As an illustration, we consider two paradigmatic qubit semigroups: Pauli dynamical maps and phase-covariant dynamics. This result has two interesting implications: it provides a universal constraint for the spectra of qubit Schwarz maps and gives rise to a necessary condition for a Schwarz qubit map to be Markovian.

arXiv:2401.05051Journal of Physics A Mathematical and TheoreticalApr 9, 20244 citations

Diagonal Unitary Covariant Superchannels

Dariusz Chruściński, Vivek Pandey, Sohail

We present a complete characterization of diagonal unitary covariant (DU-covariant) superchannels, i.e. higher-order transformations transforming quantum channels into themselves. Necessary and sufficient conditions for complete positivity and trace preservation are derived and the canonical decomposition describing DU-covariant superchannels is provided. The presented framework unifies and extends known families of covariant quantum channels and enables explicit analysis of their action on physically relevant examples, including amplitude-damping, bit-flip, and Pauli channels. Our results provide a practical toolbox for symmetry-restricted higher-order quantum processes and offer a setting for exploring open problems such as the celebrated positive partial transpose (PPT)-squared conjecture.We present a complete characterization of diagonal unitary covariant (DU-covariant) superchannels, i.e. higher-order transformations transforming quantum channels into themselves. Necessary and sufficient conditions for complete positivity and trace preservation are derived and the canonical decomposition describing DU-covariant superchannels is provided. The presented framework unifies and extends known families of covariant quantum channels and enables explicit analysis of their action on physically relevant examples, including amplitude-damping, bit-flip, and Pauli channels. Our results provide a practical toolbox for symmetry-restricted higher-order quantum processes and offer a setting for exploring open problems such as the celebrated positive partial transpose (PPT)-squared conjecture.

quant-pharXiv:2512.24389Journal of Mathematical PhysicsDec 31, 20250 citations

Diagonal unitary and orthogonal symmetries in quantum theory

Satvik Singh, Ion Nechita

We analyze bipartite matrices and linear maps between matrix algebras, which are respectively, invariant and covariant, under the diagonal unitary and orthogonal groups' actions. By presenting an expansive list of examples from the literature, which includes notable entries like the Diagonal Symmetric states and the Choi-type maps, we show that this class of matrices (and maps) encompasses a wide variety of scenarios, thereby unifying their study. We examine their linear algebraic structure and investigate different notions of positivity through their convex conic manifestations. In particular, we generalize the well-known cone of completely positive matrices to that of triplewise completely positive matrices and connect it to the separability of the relevant invariant states (or the entanglement breaking property of the corresponding quantum channels). For linear maps, we provide explicit characterizations of the stated covariance in terms of their Kraus, Stinespring, and Choi representations, and systematically analyze the usual properties of positivity, decomposability, complete positivity, and the like. We also describe the invariant subspaces of these maps and use their structure to provide necessary and sufficient conditions for separability of the associated invariant bipartite states.We analyze bipartite matrices and linear maps between matrix algebras, which are respectively, invariant and covariant, under the diagonal unitary and orthogonal groups' actions. By presenting an expansive list of examples from the literature, which includes notable entries like the Diagonal Symmetric states and the Choi-type maps, we show that this class of matrices (and maps) encompasses a wide variety of scenarios, thereby unifying their study. We examine their linear algebraic structure and investigate different notions of positivity through their convex conic manifestations. In particular, we generalize the well-known cone of completely positive matrices to that of triplewise completely positive matrices and connect it to the separability of the relevant invariant states (or the entanglement breaking property of the corresponding quantum channels). For linear maps, we provide explicit characterizations of the stated covariance in terms of their Kraus, Stinespring, and Choi representations, and systematically analyze the usual properties of positivity, decomposability, complete positivity, and the like. We also describe the invariant subspaces of these maps and use their structure to provide necessary and sufficient conditions for separability of the associated invariant bipartite states.

arXiv:2010.07898QuantumAug 9, 202127 citations

Sufficient conditions for the Kadison--Schwarz property of unital positive maps on $M_3$ Sufficient conditions for the Kadison--Schwarz property of unital positive maps on $M_3$

Adam Rutkowski

Kadison--Schwarz (KS) maps form a natural class of positive linear maps lying strictly between positivity and complete positivity. Despite their relevance in operator algebras and quantum dynamics, explicit analytic sufficient conditions for the KS property remain scarce beyond low-dimensional or highly symmetric settings. In this work we analyze unital positive linear maps on $M_3$ 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.Kadison--Schwarz (KS) maps form a natural class of positive linear maps lying strictly between positivity and complete positivity. Despite their relevance in operator algebras and quantum dynamics, explicit analytic sufficient conditions for the KS property remain scarce beyond low-dimensional or highly symmetric settings. In this work we analyze unital positive linear maps on $M_3$ 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.

quant-pharXiv:2512.18900Dec 23, 20250 citations

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.