scholarly journals Doubly stochastic matrices and the quantum channels

2021 ◽  
Vol 17 (1) ◽  
pp. 73-107
Author(s):  
H. K. Das ◽  
Md. Kaisar Ahmed
Keyword(s):  
Matrix Theory ◽  
Stochastic Matrix ◽  
Quantum Channels ◽  
Stochastic Matrices ◽  
Famous Conjecture ◽  
Frobenius Theorem ◽  
Doubly Stochastic Matrix ◽  
Birkhoff Theorem ◽  
Doubly Stochastic ◽  
Doubly Stochastic Matrices

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.

Permanents of Random Doubly Stochastic Matrices

1974 ◽  
Vol 26 (3) ◽  
pp. 600-607 ◽  
Author(s):  
R. C. Griffiths
Keyword(s):  
Symmetric Group ◽  
Stochastic Matrix ◽  
Stochastic Matrices ◽  
Doubly Stochastic Matrix ◽  
Doubly Stochastic ◽  
Survey Article ◽  
Doubly Stochastic Matrices ◽  
Image Position

The permanent of an n × n matrix A = (aij) is defined aswhere Sn is the symmetric group of order n. For a survey article on permanents the reader is referred to [2]. An unresolved conjecture due to van der Waerden states that if A is an n × n doubly stochastic matrix; then per (A) ≧ n!/nn, with equality if and only if A = Jn = (1/n).

An Elementary Proof of Johnson–Dulmage–Mendelsohn's Refinement of Birkhoff's Theorem on Doubly Stochastic Matrices

1979 ◽  
Vol 22 (1) ◽  
pp. 81-86 ◽  
Author(s):  
Akihiro Nishi
Keyword(s):  
Upper Bound ◽  
Elementary Proof ◽  
Convex Combination ◽  
Stochastic Matrix ◽  
Stochastic Matrices ◽  
Doubly Stochastic Matrix ◽  
Doubly Stochastic ◽  
Permutation Matrices ◽  
Doubly Stochastic Matrices ◽  
Birkhoff’S Theorem

SummaryA purely combinatorial and elementary proof of Johnson-Dulmage-Mendelsohn's theorem, which gives a quite sharp upper bound on the number of permutation matrices needed for representing a doubly stochastic matrix by their convex combination, is given.

scholarly journals Diagonal Sums of Doubly Substochastic Matrices

2019 ◽  
Vol 35 ◽  
pp. 42-52
Author(s):  
Lei Cao ◽  
Zhi Chen ◽  
Xuefeng Duan ◽  
Selcuk Koyuncu ◽  
Huilan Li
Keyword(s):  
Symmetric Group ◽  
Stochastic Matrix ◽  
Convex Polytope ◽  
Stochastic Matrices ◽  
Doubly Stochastic Matrix ◽  
Doubly Stochastic ◽  
Doubly Stochastic Matrices

Let $\Omega_n$ denote the convex polytope of all $n\times n$ doubly stochastic matrices, and $\omega_{n}$ denote the convex polytope of all $n\times n$ doubly substochastic matrices. For a matrix $A\in\omega_n$, define the sub-defect of $A$ to be the smallest integer $k$ such that there exists an $(n+k)\times(n+k)$ doubly stochastic matrix containing $A$ as a submatrix. Let $\omega_{n,k}$ denote the subset of $\omega_n$ which contains all doubly substochastic matrices with sub-defect $k$. For $\pi$ a permutation of symmetric group of degree $n$, the sequence of elements $a_{1\pi(1)},a_{2\pi(2)}, \ldots, a_{n\pi(n)}$ is called the diagonal of $A$ corresponding to $\pi$. Let $h(A)$ and $l(A)$ denote the maximum and minimum diagonal sums of $A\in \omega_{n,k}$, respectively. In this paper, existing results of $h$ and $l$ functions are extended from $\Omega_n$ to $\omega_{n,k}.$ In addition, an analogue of Sylvesters law of the $h$ function on $\omega_{n,k}$ is proved.

scholarly journals On an Algorithm of G. Birkhoff Concerning Doubly Stochastic Matrices

1960 ◽  
Vol 3 (3) ◽  
pp. 237-242 ◽  
Author(s):  
Diane M. Johnson ◽  
A. L. Dulmage ◽  
N. S. Mendelsohn
Keyword(s):  
Convex Hull ◽  
Upper Bound ◽  
Stochastic Matrix ◽  
Stochastic Matrices ◽  
Von Neumann ◽  
Doubly Stochastic Matrix ◽  
Doubly Stochastic ◽  
Permutation Matrices ◽  
Doubly Stochastic Matrices ◽  
Set Of Points

In [1] G. Birkhoff stated an algorithm for expressing a doubly stochastic matrix as an average of permutation matrices. In this note we prove two graphical lemmas and use these to find an upper bound for the number of permutation matrices which the Birkhoff algorithm may use.A doubly stochastic matrix is a matrix of non-negative elements with row and column sums equal to unity and is there - fore a square matrix. A permutation matrix is an n × n doubly stochastic matrix which has n2-n zeros and consequently has n ones, one in each row and one in each column. It has been shown by Birkhoff [1],Hoffman and Wielandt [5] and von Neumann [7] that the set of all doubly stochastic matrices, considered as a set of points in a space of n2 dimensions constitute the convex hull of permutation matrices.

On the Permanent of a Doubly Stochastic Matrix

1966 ◽  
Vol 18 ◽  
pp. 758-761 ◽  
Author(s):  
Herbert S. Wilf
Keyword(s):  
Stochastic Matrix ◽  
Upper Bounds ◽  
Stochastic Matrices ◽  
Doubly Stochastic Matrix ◽  
Doubly Stochastic ◽  
Doubly Stochastic Matrices ◽  
Image Position

If is an n X n matrix, the permanent of A, Per A, is defined by1where the sum is over all permutations. If A is doubly stochastic (i.e., nonnegative with row and column sums all equal to 1), then it has been conjectured that Per A ⩾ n!/nn. When confronted with a vaguely similar problem about determinants, M. Kac (1) observed that the computation of minima can often be aided by knowledge of various averages. In this spirit we calculate here the average permanent of a class of doubly stochastic matrices and thereby obtain upper bounds for the minima. These turn out to be surprisingly sharp.

A short proof of the Farahat-Mirsky refinement of Birkhoff's theorem on doubly-stochastic matrices

1961 ◽  
Vol 57 (3) ◽  
pp. 681-681 ◽  
Author(s):  
J. M. Hammersley
Keyword(s):  
Short Proof ◽  
Convex Combination ◽  
Stochastic Matrix ◽  
Stochastic Matrices ◽  
Doubly Stochastic Matrix ◽  
Doubly Stochastic ◽  
Permutation Matrices ◽  
Doubly Stochastic Matrices ◽  
Birkhoff's Theorem ◽  
Birkhoff’S Theorem

A doubly-stochastic matrix is an n × n matrix with non-negative elements such that each row and each column sums to 1. A permutation matrix is the result of permuting the rows of the unit n × n matrix. Birkhoff's theorem states that the doubly-stochastic matrices constitute the convex hull of the permutation matrices. Using Birkhoff's theorem, Farahat and Mirsky (1) showed that each doubly-stochastic matrix could be expressed as a convex combination of n2 − 2n + 2 permutation matrices, though not in general of fewer. Given Birkhoff's theorem, the Farahat-Mirsky refinement can also be proved quite shortly as follows.

A Relationship between Arbitrary Positive Matrices and Stochastic Matrices

1966 ◽  
Vol 18 ◽  
pp. 303-306 ◽  
Author(s):  
Richard Sinkhorn
Keyword(s):  
Stochastic Matrix ◽  
Diagonal Matrix ◽  
Stochastic Matrices ◽  
Doubly Stochastic Matrix ◽  
Doubly Stochastic ◽  
Positive Matrices ◽  
Square Matrix

The author (2) has shown that corresponding to each positive square matrix A (i.e. every aij > 0) is a unique doubly stochastic matrix of the form D1AD2, where the Di are diagonal matrices with positive diagonals. This doubly stochastic matrix can be obtained as the limit of the iteration defined by alternately normalizing the rows and columns of A.In this paper, it is shown that with a sacrifice of one diagonal D it is still possible to obtain a stochastic matrix. Of course, it is necessary to modify the iteration somewhat. More precisely, it is shown that corresponding to each positive square matrix A is a unique stochastic matrix of the form DAD where D is a diagonal matrix with a positive diagonal. It is shown further how this stochastic matrix can be obtained as a limit to an iteration on A.

Nonnegative generalized doubly stochastic matrices with prescribed elementary divisors

2015 ◽  
Vol 30 ◽  
pp. 704-720
Author(s):  
Ricardo Soto ◽  
Elvis Valero ◽  
Mario Salas ◽  
Hans Nina
Keyword(s):  
Stochastic Matrix ◽  
Sufficient Conditions ◽  
Stochastic Matrices ◽  
Jordan Canonical Form ◽  
Doubly Stochastic ◽  
Elementary Divisors ◽  
Doubly Stochastic Matrices ◽  
Solution Matrix ◽  
Generalized Doubly Stochastic Matrix ◽  
Perron Eigenvalue

This paper provides sufficient conditions for the existence of nonnegative generalized doubly stochastic matrices with prescribed elementary divisors. These results improve previous results and the constructive nature of their proofs allows for the computation of a solution matrix. In particular, this paper shows how to transform a generalized stochastic matrix into a nonnegative generalized doubly stochastic matrix, at the expense of increasing the Perron eigenvalue, but keeping other elementary divisors unchanged. Under certain restrictions, nonnegative generalized doubly stochastic matrices can be constructed, with spectrum \Lambda = {\lambda_1,\lambda_2 2, . . . , \lambda_n} for each Jordan canonical form associated with \Lambda

scholarly journals Eigenvalues and triangles in graphs

2020 ◽  
pp. 1-13
Author(s):  
Huiqiu Lin ◽  
Bo Ning ◽  
Baoyindureng Wu
Keyword(s):  
Bipartite Graph ◽  
Adjacency Matrix ◽  
Matrix Theory ◽  
Stochastic Matrix ◽  
Extremal Graphs ◽  
Open Problems ◽  
Free Graph ◽  
Doubly Stochastic Matrix ◽  
Doubly Stochastic ◽  
Largest Eigenvalues

Abstract Bollobás and Nikiforov (J. Combin. Theory Ser. B.97 (2007) 859–865) conjectured the following. If G is a Kr+1-free graph on at least r+1 vertices and m edges, then ${\rm{\lambda }}_1^2(G) + {\rm{\lambda }}_2^2(G) \le (r - 1)/r \cdot 2m$ , where λ1 (G)and λ2 (G) are the largest and the second largest eigenvalues of the adjacency matrix A(G), respectively. In this paper we confirm the conjecture in the case r=2, by using tools from doubly stochastic matrix theory, and also characterize all families of extremal graphs. Motivated by classic theorems due to Erdős and Nosal respectively, we prove that every non-bipartite graph G of order n and size m contains a triangle if one of the following is true: (i) ${{\rm{\lambda }}_1}(G) \ge \sqrt {m - 1} $ and $G \ne {C_5} \cup (n - 5){K_1}$ , and (ii) ${{\rm{\lambda }}_1}(G) \ge {{\rm{\lambda }}_1}(S({K_{[(n - 1)/2],[(n - 1)/2]}}))$ and $G \ne S({K_{[(n - 1)/2],[(n - 1)/2]}})$ , where $S({K_{[(n - 1)/2],[(n - 1)/2]}})$ is obtained from ${K_{[(n - 1)/2],[(n - 1)/2]}}$ by subdividing an edge. Both conditions are best possible. We conclude this paper with some open problems.

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