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

Seyed Mahmoud Manjegani, Shirin Moein

Published Oct 22, 2021• 0 citations
OpenAlex

Abstract

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.

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Characterization and Preservation of Majorization on L1(X)L^1(X)L1(X) with Some Applications in Quantum Information Theory
This work devoted to extension of majorization based on semi doubly stochastic operators(denoted by SD(L1)S\mathcal{D}(L^1)SD(L1)) as a subclass of Markov operators but a larger class of doubly stochastic operator to L1(X)L^1(X)L1(X) when XXX is a σ\sigmaσ-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∈SD(L1)}\{Sf:~~S\in S\mathcal{D}(L^1)\}{Sf:  S∈SD(L1)} when fff is a fix element in L1(X)L^1(X)L1(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.
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.
The question of how to characterize disorder arises in a wide variety of settings, including information theory, economics, and quantum thermodynamics. The theory of majorization provides an elegant answer, playing a central role in these fields. However, the existing majorization framework is inapplicable to functions that both take negative values and are defined on infinite spaces. Yet such functions, in the form of quasiprobability distributions, are ubiquitous in fields such as quantum optics, signal analysis, and bosonic quantum computation. Here we develop a notion of majorization that is applicable to such functions, proving that it admits four equivalent characterizations that naturally reduce to the finite case, thereby generalizing a seminal theorem by Hardy, Littlewood, and Pólya. Moreover, we extend this equivalence to the setting where majorization is considered relative to an arbitrary positive distribution. We give several applications of our results in the context of quantum resource theories. These include deriving families of resource monotones and constraining quantum state conversions. We analytically and numerically study examples using the Wigner and Husimi functions, which feature prominently in quantum optics. Our results provide a comprehensive majorization framework for assessing the disorder of integrable functions over infinite measure spaces. Quasiprobability distributions, crucial in quantum optics and computation, challenge existing majorization frameworks due to their negative values and infinite domains. Here, the authors establish a novel majorization concept for such functions, offering four equivalent characterizations and extending applications to quantum resource theories, enhancing our understanding of quantum state transformations.
We introduce a theory of noncommutative majorization that extends the classical majorization theory introduced by Hardy, Littlewood and Pólya to tuples of self-adjoint matrices that do not necessarily commute. We define and characterize a noncommutative majorization order that extends the classical majorization order. The definition is in terms of convex noncommutative functions, and we utilize the noncommutative convexity theory and noncommutative Choquet theory recently introduced by Davidson and the first author. As an application, we obtain a new necessary and sufficient condition for the existence of a trace-preserving completely positive map, i.e. a quantum channel, that interpolates between two finite sets of matrices. We give examples demonstrating that it is not always possible for this map to be chosen mixed unitary, even locally. We also address the computational difficulty of verifying the noncommutative majorization order. Our results further apply beyond the tracial case, and we obtain more general results characterizing the existence of unital completely positive maps that preserve an arbitrary faithful state or even unital completely positive map.
Given a positive operator-valued measure ννν acting on the Borel sets of a locally compact Hausdorff space XXX, with outcomes in the algebra B(H)\mathcal B(\mathcal H)B(H) of all bounded operators on a (possibly infinite-dimensional) Hilbert space H\mathcal HH, one can consider ννν-integrable functions X→B(H)X\rightarrow \mathcal B(\mathcal H)X→B(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.
Motivated by reachability questions in coherently controlled open quantum systems coupled to a thermal bath, as well as recent progress in the field of thermo-/vector-ddd-majorization we generalize classical majorization from unital quantum channels to channels with an arbitrary fixed point DDD of full rank. Such channels preserve some Gibbs-state and thus play an important role in the resource theory of quantum thermodynamics, in particular in thermo-majorization. Based on this we investigate DDD-majorization on matrices in terms of its topological and order properties, such as existence of unique maximal and minimal elements, etc. Moreover we characterize D-majorization in the qubit case via the trace norm and elaborate on why this is a challenging task when going beyond two dimensions.
Based on the quantum white noise theory, we formulate new types of anticipating quantum stochastic integrals by combining the Hitsuda–Skorokhod quantum stochastic integrals and the interactions between the integrands and the integrators. For our purpose, we prove various versions of analytic characterization theorems of symbols of white noise operators as more general cases. Also, several types of quantum (stochastic) gradients are formulated as continuous linear operators which are related to the quantum white noise derivatives and reflecting the interactions with the external noises. Finally, we formulate further systematical study of anticipating quantum stochastic integrals.