Semi-doubly Stochastic Operators and Majorization of Integrable Functions

Bahrami, Farid; Manjegani, Seyed Mahmoud; Moein, Shirin

doi:10.1007/s40840-020-00971-2

Cited by 3 publications

(6 citation statements)

References 8 publications

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“…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.…”

Section: Introductionmentioning

confidence: 85%

“…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].…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

“…(1) It is worth to refer to an important result by Bahrami and et.al [2,Proposition 2.6] that if µ(X) < ∞ then on L 1 (X, µ) the semi doubly stochastic operator coincide the doubly stochastic operator. We provide a counterexample which asserts that in general we no longer have this conformity and the converse of inclusions (1) are not true.…”

Section: Introductionmentioning

confidence: 99%

“…The most important reason is extending the Ryff's characterization to L 1 (X, µ) when (X, µ) is σ-finite measure space by using the semi doubly stochastic operators[2, Theorem 2.3 and Theorem 2.4]. And the other reason of significant of semi doubly stochastic operators which we can point them out from [2] is the definition of semi doubly stochastic operator is independent of decreasing rearrangement but majorization based on decreasing rearrangement i.e. g ≺ f in form of definition 2 are strongly related to the majorization based on semi doubly stochastic operators i.e.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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

Manjegani¹,

Moein²

2021

Preprint

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This work devoted to extension of majorization based on semi doubly stochastic operators(denoted by SD(L 1 )) as a subclass of Markov operators but a larger class of doubly stochastic operator to L 1 (X) when X is a σ-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 [6] and answer affirmatively to Mirsky's question [11] about extension of doubly stochastic matrix to infinite dimension as an infinite continuous counterpart of Hardy, Littlewood and Pólya's characterization of majorization. We collect a number of results of semi doubly stochastic operators such as equi-integrability of {Sf : S ∈ SD(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.

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Section: Introductionmentioning

confidence: 85%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Manjegani¹,

Moein²

2021

Preprint

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Majorization and semidoubly stochastic operators on $L^{1}(X)$

Manjegani

Moein

2023

J Inequal Appl

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This paper is devoted to a study of majorization based on semidoubly stochastic operators (denoted by $S\mathcal{D}(L^{1})$ S D ( L 1 ) ) on $L^{1}(X)$ L 1 ( X ) when X is a σ-finite measure space. We answer Mirsky’s question and characterize the majorization by means of semidoubly stochastic maps on $L^{1}(X)$ L 1 ( X ) . We prove some results on semidoubly stochastic operators such as a strong relation of semidoubly stochastic operators and integral stochastic operators and relatively weak compactness of $S_{f}=\{Sf : S\in S\mathcal{D}(L^{1})\}$ S f = { S f : S ∈ S D ( L 1 ) } for a fixed element $f\in L^{1}(X)$ f ∈ L 1 ( X ) by proving the equiintegrability of $S_{f}$ S f . We present a full characterization of majorization on a σ-finite measure space X.

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Continuous majorization in quantum phase space

Herstraeten

Jabbour

Cerf

2023

Quantum

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We explore the role of majorization theory in quantum phase space. To this purpose, we restrict ourselves to quantum states with positive Wigner functions and show that the continuous version of majorization theory provides an elegant and very natural approach to exploring the information-theoretic properties of Wigner functions in phase space. After identifying all Gaussian pure states as equivalent in the precise sense of continuous majorization, which can be understood in light of Hudson's theorem, we conjecture a fundamental majorization relation: any positive Wigner function is majorized by the Wigner function of a Gaussian pure state (especially, the bosonic vacuum state or ground state of the harmonic oscillator). As a consequence, any Schur-concave function of the Wigner function is lower bounded by the value it takes for the vacuum state. This implies in turn that the Wigner entropy is lower bounded by its value for the vacuum state, while the converse is notably not true. Our main result is then to prove this fundamental majorization relation for a relevant subset of Wigner-positive quantum states which are mixtures of the three lowest eigenstates of the harmonic oscillator. Beyond that, the conjecture is also supported by numerical evidence. We conclude by discussing some implications of this conjecture in the context of entropic uncertainty relations in phase space.

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Semi-doubly Stochastic Operators and Majorization of Integrable Functions

Cited by 3 publications

References 8 publications

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

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

Majorization and semidoubly stochastic operators on $L^{1}(X)$

Continuous majorization in quantum phase space

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