Necessary and Sufficient Conditions for the Trumping Relation

Abstract

Entanglement catalysis allows one to convert certain entangled states into others by the temporary involvement of another entangled state (so-called catalyst), where after the conversion the catalyst is returned to the same state. For bipartite pure entangled states that can be transformed in this way with unit probability, the respective Schmidt coefficients are said to satisfy the trumping relation, a mathematical relation which is an extension of the majorization relation. This article provides all necessary and sufficient conditions for the trumping relation in terms of the Schmidt coefficients. The coefficients should satisfy strict inequalities for the entropy of entanglement and for power means excluding the special power 1.

9 pages; comments appreciated; (this article is a corrected and modified version, which combines this preprint with arXiv:0706.3654)

References in corpus (10)

• Concentrating Partial Entanglement by Local Operations
• Conditions for a class of entanglement transformations
• Entanglement monotones
• Entanglement of pure states for a single copy
• Entanglement-assisted local manipulation of pure quantum states
• The trumping relation and the structure of the bipartite entangled states
• Multiple-copy entanglement transformation and entanglement catalysis
• A classification of incomparable states
• Catalyst-assisted Probabilistic Entanglement Transformation
• When Catalysis is Useful for Probabilistic Entanglement Transformation

Cited by in corpus (77)

• Quantum Resource Theories
• The role of quantum information in thermodynamics --- a topical review
• The second laws of quantum thermodynamics
• The resource theory of informational nonequilibrium in thermodynamics
• An introductory review of the resource theory approach to thermodynamics
• Operational one-to-one mapping between coherence and entanglement measures
• Entanglement negativity and entropy in non-equilibrium conformal field theory
• Correlating thermal machines and the second law at the nanoscale
• Entanglement Entropy of Non Unitary Conformal Field Theory
• Von Neumann entropy from unitarity
• Stochastic independence as a resource in small-scale thermodynamics
• Quantum phases with differing computational power
• Catalytic quantum teleportation and beyond
• Quantum thermodynamics of correlated-catalytic state conversion at small-scale
• Catalytic coherence transformations
• Catalytic Transformations of Pure Entangled States
• Catalysis of entanglement and other quantum resources
• Correlation in Catalysts Enables Arbitrary Manipulation of Quantum Coherence
• Entropy and reversible catalysis
• Entanglement Rényi $\alpha$-entropy
• A generalization of majorization that characterizes Shannon entropy
• Fundamental Limits on Correlated Catalytic State Transformations
• Entropy and relative entropy from information-theoretic principles
• Local characterization of 1d topologically ordered states
• All states are universal catalysts in quantum thermodynamics
• Catalyst-Assisted Probabilistic Coherence Distillation for Mixed States
• Local convertibility and the quantum simulation of edge states in many-body systems
• Necessary conditions on entanglement catalysts
• Local response of topological order to an external perturbation
• Relative Entropy and Catalytic Relative Majorization
• Entanglement convertibility by sweeping through the quantum phases of the alternating bonds $XXZ$ chain
• Amplifying asymmetry with correlated catalysts
• Fundamental limits on anomalous energy flows in correlated quantum systems
• One-shot Coherence Distillation with Catalysts
• Extending a characterization of majorization to infinite dimensions
• Arbitrary Amplification of Quantum Coherence in Asymptotic and Catalytic Transformation
• Approximate transformations of bipartite pure-state entanglement from the majorization lattice
• Autonomous Quantum Devices: When Are They Realizable without Additional Thermodynamic Costs?
• Local convertibility of the ground state of the perturbed Toric code
• Catalytic Transformations in Coherence Theory
• Two constructive proofs on d-majorization and thermo-majorization
• Non universality of entanglement convertibility
• The minimum entropy output of a quantum channel is locally additive
• Stochastic domination for iterated convolutions and catalytic majorization
• Quantum Foundations of Classical Reversible Computing
• Entanglement-assisted local operations and classical communications conversion in the quantum critical systems
• No-go theorem for entanglement distillation using catalysis
• Entanglement catalysis for quantum states and noisy channels
• Bounds On Entanglement Catalysts
• Local Transformations of Multiple Multipartite States
• Geometric relative entropies and barycentric Rényi divergences
• The classification of preordered spaces in terms of monotones: complexity and optimization
• Quantum catalysis in cavity quantum electrodynamics
• Entanglement fluctuation theorems
• Catalytic transformations for thermal operations
• Matrix majorization in large samples
• Teleportation with Embezzling Catalysts
• Representing preorders with injective monotones
• Dirichlet polynomials, Majorization, and Trumping
• Breakdown of local convertibility through Majorana modes in a quantum quench
• Necessary conditions on effective quantum entanglement catalysts
• Phase diagram of quantum critical system via local convertibility of ground state
• Quantum thermodynamics with coherence: Covariant Gibbs-preserving operation is characterized by the free energy
• Communication with Quantum Catalysts
• Catalytic transformations with finite-size environments: applications to cooling and thermometry
• Single-shot entanglement manipulation of states and channels revisited
• The minimum Renyi entropy output of a quantum channel is locally additive
• A generalized dominance ordering for 1/2-BPS states
• Majorization requires infinitely many second laws
• Approximate and ensemble local entanglement transformations for multipartite states
• Catalytic Activation of Bell Nonlocality
• Unlocking inaccessible performance of the quantum refrigerator with catalysts
• A finite sufficient set of conditions for catalytic majorization
• Classical noise and the structure of minimal uncertainty states
• Symmetric polynomials and $l^p$ inequalities for certain intervals of $p$
• Finite-size catalysis in quantum resource theories
• Trumping and Power Majorization