<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>The main object of this paper is to study doubly stochastic matrices with majorization and the Birkhoff theorem. The Perron-Frobenius theorem on eigenvalues is generalized for doubly stochastic matrices. The region of all possible eigenvalues of n-by-n doubly stochastic matrix is the union of regular (n – 1) polygons into the complex plane. This statement is ensured by a famous conjecture known as the Perfect-Mirsky conjecture which is true for n = 1, 2, 3, 4 and untrue for n = 5. We show the extremal eigenvalues of the Perfect-Mirsky regions graphically for n = 1, 2, 3, 4 and identify corresponding doubly stochastic matrices. Bearing in mind the counterexample of Rivard-Mashreghi given in 2007, we introduce a more general counterexample to the conjecture for n = 5. Later, we discuss different types of positive maps relevant to Quantum Channels (QCs) and finally introduce a theorem to determine whether a QCs gives rise to a doubly stochastic matrix or not. This evidence is straightforward and uses the basic tools of matrix theory and functional analysis.</p>
</abstract>

The main object of this paper is to study doubly stochastic matrices with majorization and the Birkhoff theorem. The Perron-Frobenius theorem on eigenvalues is generalized for doubly stochastic matrices. The region of all possible eigenvalues of n-by-n doubly stochastic matrix is the union of regular (n – 1) polygons into the complex plane. This statement is ensured by a famous conjecture known as the Perfect-Mirsky conjecture which is true for n = 1, 2, 3, 4 and untrue for n = 5. We show the extremal eigenvalues of the Perfect-Mirsky regions graphically for n = 1, 2, 3, 4 and identify corresponding doubly stochastic matrices. Bearing in mind the counterexample of Rivard-Mashreghi given in 2007, we introduce a more general counterexample to the conjecture for n = 5. Later, we discuss different types of positive maps relevant to Quantum Channels (QCs) and finally introduce a theorem to determine whether a QCs gives rise to a doubly stochastic matrix or not. This evidence is straightforward and uses the basic tools of matrix theory and functional analysis.

Doubly stochastic matrices and the quantum channels

Journal of Applied Mathematics, Statistics and Informatics

This work is licensed under the Creative Commons Attribution 4.0 International License.

{"article-title":"Doubly stochastic matrices and the quantum channels"}

The main object of this paper is to study doubly stochastic matrices with majorization and the Birkhoff theorem. The Perron-Frobenius theorem on eigenvalues is...

Doubly stochastic matrices and the quantum channels

Publié en ligne: 17 juin 2021

Pages: 73 - 107

DOI: https://doi.org/10.2478/jamsi-2021-0005

Mots clés
Stochastic Matrices, Doubly Stochastic Matrices, Eigenvalues, Eigenvectors, Linear Operators, Majoriation, Perfect-Mirsky Conjecture, Quantum Channel

© 2021 H. K. Das et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Doubly stochastic matrices and the quantum channels

Publié en ligne: 17 juin 2021

Pages: 73 - 107

DOI: https://doi.org/10.2478/jamsi-2021-0005

Mots clésStochastic Matrices, Doubly Stochastic Matrices, Eigenvalues, Eigenvectors, Linear Operators, Majoriation, Perfect-Mirsky Conjecture, Quantum Channel

© 2021 H. K. Das et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Mots clés
Stochastic Matrices, Doubly Stochastic Matrices, Eigenvalues, Eigenvectors, Linear Operators, Majoriation, Perfect-Mirsky Conjecture, Quantum Channel