Extremal distributions under approximate majorization

Horodecki, Michał; Oppenheim, Jonathan; Sparaciari, Carlo

doi:10.1088/1751-8121/aac87c

Cited by 22 publications

(38 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In addition, we showed that the notion of approximate majorization is in strong connection with the property of completeness of the majorization lattice [20,42]. Indeed, the flattest and steepest approximations are nothing more than the infimum and supremum of the corresponding set, respectively, and they can be calculated only from their vertices.…”

Section: Discussionmentioning

confidence: 91%

“…, of x 0 given in [20,42] , although the algorithms to obtain them are different to the ones presented here. Thus, we see that the notion of approximate majorization is in strong connection with the property of completeness of the majorization lattice.…”

Section: Infimum and Supremum Over Convex Polytopesmentioning

confidence: 95%

“…Therefore, by applying lemma 1, B x 0  ⋀ ( )and B x 0  ⋁ ( ) reduces to finding the infimum and supremum of the vertices of B x 0  ( ). Interestingly enough, the lattice-theoretic property of majorization can be posed in strong connection with the notion of approximate majorization [20,42], which has recently found application in quantum thermodynamics [43]. More precisely, the steepest ò-approximation,…”

Section: Infimum and Supremum Over Convex Polytopesmentioning

confidence: 99%

See 2 more Smart Citations

Optimal common resource in majorization-based resource theories

et al. 2019

View full text Add to dashboard Cite

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 nondenumerable 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.

show abstract

Section: Discussionmentioning

confidence: 91%

Section: Infimum and Supremum Over Convex Polytopesmentioning

confidence: 95%

Section: Infimum and Supremum Over Convex Polytopesmentioning

confidence: 99%

See 1 more Smart Citation

Optimal common resource in majorization-based resource theories

et al. 2019

View full text Add to dashboard Cite

show abstract

“…This characterization is not useful enough to obtain the supremum and infimum, since one needs to know a priori the vertices of B p (x). Notwithstanding, for p = 1, it has recently been shown not only how to compute supremum and infimum, but also that they are the maximum and minimum of the ball B 1 (x) [18,31]. Here, we complete the order-theoretic characterization of the balls by analyzing the case p = ∞, showing that B ∞ (x) also admits extremal probability vectors.…”

Section: Order-theoretic Properties Ofmentioning

confidence: 86%

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

Massri¹,

Bellomo²,

Holik³

et al. 2020

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

Given a probability vector x with its components sorted in non-increasing order, we consider the closed ball B p (x) with p ≥ 1 formed by the probability vectors whose p -norm distance to the center x is less than or equal to a radius . 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 p (x) with 1 < p < ∞. On the other hand, we show that B ∞ (x) is a complete sublattice of the majorization lattice. As a consequence, this ball has also 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 1 -norm. Finally, we apply our results to the problem of approximate conversion of resources within the framework of quantum resource theory of nonuniformity.

show abstract

“…This majorization-based UR provides universal applicability to any appropriate uncertainty functions with such an uncertainty-order preserving property. Besides uncertainty relations, the concept of majorization is applied to various topics, such as quantum thermodynamics [31] and coherence [32].…”

Section: Introductionmentioning

confidence: 99%

Entropic Uncertainty Relations via Direct-Sum Majorization Relation for Generalized Measurements

Baek

Nha

Son

2019

Entropy

View full text Add to dashboard Cite

We derive an entropic uncertainty relation for generalized positive-operator-valued measure (POVM) measurements via a direct-sum majorization relation using Schur concavity of entropic quantities in a finite-dimensional Hilbert space. Our approach provides a significant improvement of the uncertainty bound compared with previous majorization-based approaches [S. Friendland, V. Gheorghiu and G. Gour, Phys. Rev. Lett. 111, 230401 (2013); A. E. Rastegin and K.Życzkowski, J. Phys. A, 49, 355301 (2016)], particularly by extending the direct-sum majorization relation first introduced in [ L. Rudnicki, Z. Pucha la and K.Życzkowski, Phys. Rev. A 89, 052115 (2014)]. We illustrate the usefulness of our uncertainty relations by considering a pair of qubit observables in a two-dimensional system and randomly chosen unsharp observables in a three-dimensional system. We also demonstrate that our bound tends to be stronger than the generalized Maassen-Uffink bound with an increase in the unsharpness effect. Furthermore, we extend our approach to the case of multiple POVM measurements, thus making it possible to establish entropic uncertainty relations involving more than two observables.

show abstract

Extremal distributions under approximate majorization

Cited by 22 publications

References 24 publications

Optimal common resource in majorization-based resource theories

Optimal common resource in majorization-based resource theories

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

Entropic Uncertainty Relations via Direct-Sum Majorization Relation for Generalized Measurements

Contact Info

Product

Resources

About