Quasi doubly stochastic operator on l1 and Nielsen’s theorem

Abstract

In this paper, we introduce the quasidoubly stochastic operator, which is between doubly stochastic operators and column stochastic operators, so as to apply to characterized operator S on l1 such that Sf is majorized by f for every f ∈ l1. We present some classes of majorization preservers on l1 under quasi doubly stochastic operators. Moreover, as an application of our result in quantum physics, the convertibility of pure states of a composite system by local operations and classical communication has been considered.

Majorization and Semi-Doubly Stochastic Operators on $L^1(X)$

“…Recently, we introduced semi doubly stochastic operators as an important class of operators on L 1 (X) when X is σfinite measure space [2,10]. In this work, we claim that semi doubly stochastic operator is more suitable for extending the doubly stochastic matrix to σ-finite measure space and also completely answer Mirsky's question(extension of Hardy, Littlewood and Pólya's results) and refuse Hiai's conjecture based on semi doubly stochastic operators.…”

“…Until recent decades, the main attention in majorization theory was paid to finite dimensional space, but recently because of its significant applications in a broad spectrum of fields, especially in quantum physics, considerable interest to infinite dimensional spaces appeared mathematically and physically [2,7,9,10,16].…”

“…So in the finite case of Nielsen's Theorem, the majorization relation is between the Schmidt vectors, and therefore based on infinite dimensional version of the Schmidt decomposition Theorem for extending the Nielsen's Theorem to infinite dimensional case we are dealing with the Schmidt sequence, instead of the Schmidt vector, with positive terms and summation over all terms unit, which is obviously belongs to l 1 space. That is why l 1 space has the best quantum physical descriptions of infinite quantum states what is discussed in [10] and [15].…”

“…Recently quasidoubly stochastic operators as a new class of operators introduced by Manjegani and Moein [10] on l 1 space and extended by Bahrami and et.al [2] to semi doubly stochastic operators on L 1 (X, µ) where (X, µ) is σ-finite measure space.…”

Submajorization on $\ell^p(I)^+$ determined by increasable doubly substochastic operators and its linear preservers

“…In recent years, there is a big progress towards developing various extensions of the most important majorization relations on sequence spaces [5,15] and on descrete Lebesgue spaces [6,7,8,9,10,18,19,20,21,22,23] with apropriate generalizations of some famous theorems in linear algebra [2,3,4,16,24,27,30,32]. There are a lot of applications of the majorization theory in various branches of mathematics and there exist significant conections with the other science like physics, quantum mechanics and quantum information theory [12,17,25,30,31,33].…”