Journal of Mathematical Physics

Quasi doubly stochastic operator on l1 and Nielsen’s theorem

Seyed Mahmoud Manjegani, Shirin Moein

Published Oct 1, 2019• 4 citations

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.

DOI: 10.1063/1.5093278

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Journal of Mathematical Physics

Quasi doubly stochastic operator on l1 and Nielsen’s theorem

Seyed Mahmoud Manjegani, Shirin Moein

Published Oct 1, 2019• 4 citations

Publisher

Abstract

DOI: 10.1063/1.5093278

Related papers

Characterization and Preservation of Majorization on $L^1(X)$ with Some Applications in Quantum Information TheoryCharacterization and Preservation of Majorization on $L^1(X)$ with Some Applications in Quantum Information Theory

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Characterization and Preservation of Majorization on

L^1(X)

with Some Applications in Quantum Information Theory

This work devoted to extension of majorization based on semi doubly stochastic operators(denoted by

S\mathcal{D}(L^1)

) as a subclass of Markov operators but a larger class of doubly stochastic operator to

L^1(X)

when

X

is a

\sigma

-finite measure space. We fully describe why such as these operators are more suitable for generalizing majorization relation. By using semi doubly stochastic operators we refuse Hiai's conjecture\cite{Hiai} and answer affirmatively to Mirsky's question\cite{Mirsky} about extension of doubly stochastic matrix to infinite dimension as an infinite continuous counterpart of Hardy, Littlewood and P{\'o}lya's characterization of majorization. We collect a number of results of semi doubly stochastic operators such as equi-integrability of

\{Sf:~~S\in S\mathcal{D}(L^1)\}

when

f

is a fix element in

L^1(X)

and consequently relatively weakly compact, and also strong relation of semi doubly stochastic operators and integral stochastic operators. Motivation of our work is application in quantum information theory like Nielson's Theorem, Uhlmann's Theorem and convertibility via incoherent operation. We explain extension of Uhlmann's Theorem to infinite dimension by means of semi doubly stochastic operators which leads to a quantum interpretation of majorization preserver. We introduce three types of maps which preserve majorization.

Given a positive operator-valued measure

ν

acting on the Borel sets of a locally compact Hausdorff space

X

, with outcomes in the algebra

\mathcal B(\mathcal H)

of all bounded operators on a (possibly infinite-dimensional) Hilbert space

\mathcal H

, one can consider

ν

-integrable functions

X\rightarrow \mathcal B(\mathcal H)

that are positive quantum random variables. We define a seminorm on the span of such functions which in the quotient leads to a Banach space. We consider bistochastic operators acting on this space and majorization of quantum random variables is then defined with respect to these operators. As in classical majorization theory, we relate majorization in this context to an inequality involving all possible convex functions of a certain type. Unlike the classical setting, continuity and convergence issues arise throughout the work.