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

Abstract

Given a probability vector $x$ with its components sorted in non-increasing order, we consider the closed ball ${\mathcal{B}}_{\epsilon }^p(x)$ with $p\ge 1$ formed by the probability vectors whose ${\ell }^p$-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 ${\mathcal{B}}_{\epsilon }^p(x)$ with $1<p<\infty$. On the other hand, we show that ${\mathcal{B}}_{\epsilon }^{\infty }(x)$ 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 ${\ell }^1$-norm. Finally, we apply our results to the problem of approximate conversion of resources within the framework of quantum resource theory of nonuniformity.

arXiv comment26 pages, 1 figure

References in corpus (21)

• Quantum Resource Theories
• Conditions for a class of entanglement transformations
• The resource theory of informational nonequilibrium in thermodynamics
• A mathematical theory of resources
• Conditions for coherence transformations under incoherent operations
• Maximal Coherence and the Resource Theory of Purity
• Operational one-to-one mapping between coherence and entanglement measures
• Contractivity of positive and trace preserving maps under $L_p$ norms
• Approximate transformations and robust manipulation of bipartite pure state entanglement
• Majorization Formulation of Uncertainty in Quantum Mechanics
• Deterministic Coherence Distillation
• Resource convertibility and ordered commutative monoids
• Beyond the thermodynamic limit: finite-size corrections to state interconversion rates
• The Optimal Uncertainty Relation
• Detecting coherence via spectrum estimation
• Approximate Majorization
• Maximum and minimum entropy states yielding local continuity bounds
• Optimal common resource in majorization-based resource theories
• Structure of the thermodynamic arrow of time in classical and quantum theories
• Approximate transformations of bipartite pure-state entanglement from the majorization lattice
• Correlative Capacity of Composite Quantum States

Cited by in corpus (2)

• Generalized coherence vector applied to coherence transformations and quantifiers
• Probabilistic pure state conversion on the majorization lattice