Doubly stochastic matrices and the quantum channels

H. K. Das,Md. Kaisar Ahmed

doi:10.2478/jamsi-2021-0005

Abstract

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.

Highlights

Stochastic and doubly stochastic matrices are mostly studied matrices for many years
Perfect et al [23] considered the same problem for double stochastic matrices, i.e., they tried to characterize the region ωn ⊂ C containing all the eigenvalues of nby-n doubly stochastic matrices
We studied the relationship between doubly stochastic matrices and majorization along with the structures of marginal doubly stochastic matrices in the Perfect-Mirsky region

Summary

INTRODUCTION

Stochastic and doubly stochastic matrices are mostly studied matrices for many years. In 1946, Dmitriev et al [9] tried to mark the region, denoted by Θn, given by the subset of the complex plane containing all possible eigenvalues of all n-by-n stochastic matrices They managed to find that area in part. Perfect et al [23] considered the same problem for double stochastic matrices, i.e., they tried to characterize the region ωn ⊂ C containing all the eigenvalues of nby-n doubly stochastic matrices. They gave a conjecture about the possible region known as the Perfect-Mirsky conjecture. Section five is devoted to the quantum channels and discusses the connection between doubly stochastic matrices and trace preserving initial positive maps. We generalize the condition, whether a quantum channel gives rise to doubly stochastic matrix or not by a theorem

MATHEMATICAL BACKGROUND

REGIONS OF CHARACTERISTIC ROOTS OF DOUBLY STOCHASTIC MATRICES

The Region ω3

The Region ω4

QUANTUM CHANNELS

CONCLUSION