Inequalities that Collectively Completely Characterize the Catalytic Majorization Relation

Abstract

For probability vectors x and y, the catalytic majorization relation x prec_T y is defined to hold when there exists a probability vector z such that x otimes z is majorized by y otimes z. In this paper, an infinite family of functions is given such that, subject to some trivial restrictions, x prec_T y if and only if f_r(x) < f_r(y) for all functions f_r in the family. An outline of a proof of this result is provided. The catalytic majorization relation is known to provide a determination of which transformations of jointly held pure quantum states are possible using local operations and classical communication when an additional jointly held state may be specified to facilitate the transformation without being consumed.

25 pages

References in corpus (1)

• Some Open Problems in Quantum Information Theory

Cited by in corpus (26)

• Description of quantum coherence in thermodynamic processes requires constraints beyond free energy
• The second laws of quantum thermodynamics
• Quantum phases with differing computational power
• Quantum thermodynamics of correlated-catalytic state conversion at small-scale
• Catalytic Transformations of Pure Entangled States
• Catalysis of entanglement and other quantum resources
• Entropy and reversible catalysis
• Catalytic majorization and ${\ell }_p$ norms
• A generalization of majorization that characterizes Shannon entropy
• Fundamental Limits on Correlated Catalytic State Transformations
• Entropy and relative entropy from information-theoretic principles
• Local convertibility and the quantum simulation of edge states in many-body systems
• Necessary conditions on entanglement catalysts
• Entanglement convertibility by sweeping through the quantum phases of the alternating bonds $XXZ$ chain
• Pure-state transformations and catalysis under operations that completely preserve positivity of partial transpose
• One-shot Coherence Distillation with Catalysts
• Extending a characterization of majorization to infinite dimensions
• Approximate transformations of bipartite pure-state entanglement from the majorization lattice
• Catalysis in Action via Elementary Thermal Operations
• Entanglement fluctuation theorems
• Dirichlet polynomials, Majorization, and Trumping
• Matrix majorization in large samples
• Catalytic transformations with finite-size environments: applications to cooling and thermometry
• Majorization requires infinitely many second laws
• Symmetric polynomials and $l^p$ inequalities for certain intervals of $p$
• Characterizing Incomparability in Quantum Information