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Extremal elements of a sublattice of the majorization lattice and approximate majorization

Massri, Cesar Dario, Bellomo, Guido, Federico Holik, Bosyk, Gustavo Martin

Published Jan 1, 2020• 5 citations
OpenAlex

Abstract

Given a probability vector x with its components sorted in non-increasing order, we consider the closed ball Bpε(x) with p ⩾ 1 formed by the probability vectors whose ℓp-norm distance to the center x is less than or equal to a radius epsilon. Here, we provide an order-theoretic characterization of these balls by using the majorization partial order. Unlike the case p = 1 previously discussed in the literature, we find that the extremal probability vectors, in general, do not exist for the closed balls Bpε(x) with 1 < p < ∞. On the other hand, we show that B∞ε(x) is a complete sublattice of the majorization lattice. As a consequence, this ball also has extremal elements. In addition, we give an explicit characterization of those extremal elements in terms of the radius and the center of the ball. This allows us to introduce some notions of approximate majorization and discuss its relation with previous results of approximate majorization given in terms of the ℓ1-norm. Finally, we apply our results to the problem of approximate conversion of resources within the framework of quantum resource theory of nonuniformity.

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Given a probability vector x with its components sorted in non-increasing order, we consider the closed ball Bpε(x) with p ⩾ 1 formed by the probability vectors whose ℓp-norm distance to the center x is less than or equal to a radius epsilon. Here, we provide an order-theoretic characterization of these balls by using the majorization partial order. Unlike the case p = 1 previously discussed in the literature, we find that the extremal probability vectors, in general, do not exist for the closed balls Bpε(x) with 1 < p < ∞. On the other hand, we show that B∞ε(x) is a complete sublattice of the majorization lattice. As a consequence, this ball also has extremal elements. In addition, we give an explicit characterization of those extremal elements in terms of the radius and the center of the ball. This allows us to introduce some notions of approximate majorization and discuss its relation with previous results of approximate majorization given in terms of the ℓ1-norm. Finally, we apply our results to the problem of approximate conversion of resources within the framework of quantum resource theory of nonuniformity.
Abstract Given a probability vector x with its components sorted in non-increasing order, we consider the closed ball B ϵ p ( x ) with p ⩾ 1 formed by the probability vectors whose ℓ p -norm distance to the center x is less than or equal to a radius ϵ . Here, we provide an order-theoretic characterization of these balls by using the majorization partial order. Unlike the case p = 1 previously discussed in the literature, we find that the extremal probability vectors, in general, do not exist for the closed balls B ϵ p ( x ) with 1 < p < ∞. On the other hand, we show that B ϵ ∞ ( x ) is a complete sublattice of the majorization lattice. As a consequence, this ball also has extremal elements. In addition, we give an explicit characterization of those extremal elements in terms of the radius and the center of the ball. This allows us to introduce some notions of approximate majorization and discuss its relation with previous results of approximate majorization given in terms of the ℓ 1 -norm. Finally, we apply our results to the problem of approximate conversion of resources within the framework of quantum resource theory of nonuniformity.
Abstract We address the problem of finding the optimal common resource for an arbitrary family of target states in quantum resource theories based on majorization, that is, theories whose conversion law between resources is determined by a majorization relationship, such as it happens with entanglement, coherence or purity. We provide a conclusive answer to this problem by appealing to the completeness property of the majorization lattice. We give a proof of this property that relies heavily on the more geometric construction provided by the Lorenz curves, which allows to explicitly obtain the corresponding infimum and supremum. Our framework includes the case of possibly non-denumerable sets of target states (i.e. targets sets described by continuous parameters). In addition, we show that a notion of approximate majorization, which has recently found application in quantum thermodynamics, is in close relation with the completeness of this lattice. Finally, we provide some examples of optimal common resources within the resource theory of quantum coherence.
We introduce and study a generalization of majorization called relative submajorization and show that it has many applications to the resource theories of thermodynamics, bipartite entanglement, and quantum coherence. Relative majorization is an ordering on pairs of vectors induced by stochastic transformations: One pair of vectors relatively majorizes another when there exists a stochastic transformation taking the former to the latter. Relative submajorization is a weakened version in which a substochastic matrix need only generate a vector pair obeying certain positivity conditions relative to the input pair. In the context of resource theories, we show that the relative submajorization characterizes both the probability and approximation error that can be obtained when transforming one resource to another, also when assisted by additional standard resources such as useful work or maximally entangled states. These characterizations have a geometric formulation as the ratios or differences, respectively, between the Lorenz curves associated with the input and output resources, making them efficient to compute. We also find several interesting bounds on the reversibility of a given transformation in terms of the properties of the forward transformation. The main technical tool used to establish these results is the linear programming duality, which is used to show that any instance of relative submajorization can be “dilated” to an instance of strict relative majorization.
Optimal approximate state conversion is a fundamental task in quantum resource theories, yet existing solutions are largely tailored to specific distinguishability measures and resource theories. We show that every optimal approximate-conversion problem in the three paradigmatic majorization-based quantum resource theories of non-uniformity, bipartite pure-state entanglement, and pure-state coherence reduces exactly to one of only four elementary taut-string constructions. This holds for problems generated by Schur-convex spectral distances and by a broad family of quantum divergences satisfying a natural reduction property with respect to the corresponding equivalence group. The resulting framework encompasses a broad class of physically relevant distinguishability measures, including the trace distance, all unitarily invariant norms, quantum relative entropy, maximal quantum fff-divergences, and the ααα-zzz Rényi relative entropies. The increment vector of the corresponding taut string completely determines both the optimal approximation error and a canonical optimal state. Our results establish a common geometric framework for approximate quantum resource conversion, revealing that a broad class of seemingly different optimization problems is governed by the same taut-string geometric principle.
The transformation of an initial bipartite pure state into a target one by means of local operations and classical communication and entangled-assisted by a catalyst defines a partial order between probability vectors. This partial order, so-called trumping majorization, is based on tensor products and the majorization relation. Here, we aim to study order properties of trumping majorization. We show that the trumping majorization partial order is indeed a lattice for four dimensional probability vectors and two dimensional catalysts. In addition, we show that the subadditivity and supermodularity of the Shannon entropy on the majorization lattice are inherited by the trumping majorization lattice. Finally, we provide a suitable definition of distance for four dimensional probability vectors.
Abstract Although an input distribution may not majorize a target distribution, it may majorize a distribution which is close to the target. Here we consider a notion of approximate majorization. For any distribution, and given a distance δ , we find the approximate distributions which majorize (are majorized by) all other distributions within the distance δ . We call these the steepest and flattest approximation. This enables one to compute how close one can get to a given target distribution under a process governed by majorization. We show that the flattest and steepest approximations preserve ordering under majorization. Furthermore, we give a notion of majorization distance. This has applications ranging from thermodynamics, entanglement theory, and economics.