Relative Entropy and Catalytic Relative Majorization

Abstract

Given two pairs of quantum states, a fundamental question in the resource theory of asymmetric distinguishability is to determine whether there exists a quantum channel converting one pair to the other. In this work, we reframe this question in such a way that a catalyst can be used to help perform the transformation, with the only constraint on the catalyst being that its reduced state is returned unchanged, so that it can be used again to assist a future transformation. What we find here, for the special case in which the states in a given pair are commuting, and thus quasi-classical, is that this catalytic transformation can be performed if and only if the relative entropy of one pair of states is larger than that of the other pair. This result endows the relative entropy with a fundamental operational meaning that goes beyond its traditional interpretation in the setting of independent and identical resources. Our finding thus has an immediate application and interpretation in the resource theory of asymmetric distinguishability, and we expect it to find application in other domains.

arXiv commentv3: 18 pages, 1 figure, accepted for publication in Physical Review Research

References in corpus (26)

• Concentrating Partial Entanglement by Local Operations
• Rényi Divergence and Kullback-Leibler Divergence
• Quantum Resource Theories
• Conditions for a class of entanglement transformations
• Operational Resource Theory of Coherence
• Fundamental limitations for quantum and nano thermodynamics
• The second laws of quantum thermodynamics
• The Resource Theory of Quantum States Out of Thermal Equilibrium
• Quantum majorization and a complete set of entropic conditions for quantum thermodynamics
• Entanglement-assisted local manipulation of pure quantum states
• Reversible transformations from pure to mixed states, and the unique measure of information
• Conditions for coherence transformations under incoherent operations
• Operational one-to-one mapping between coherence and entanglement measures
• Catalytic Conversion Probabilities for Bipartite Pure States
• Necessary and Sufficient Conditions for the Trumping Relation
• Correlating thermal machines and the second law at the nanoscale
• Von Neumann entropy from unitarity
• The trumping relation and the structure of the bipartite entangled states
• Quantum Relative Lorenz Curves
• Catalytic coherence transformations
• Relative submajorization and its use in quantum resource theories
• Resource theory of asymmetric distinguishability
• An information-theoretic treatment of quantum dichotomies
• Avoiding irreversibility: engineering resonant conversions of quantum resources
• One-shot Coherence Distillation with Catalysts
• Superadditivity of quantum relative entropy for general states

Cited by in corpus (21)

• Quantum thermodynamics of correlated-catalytic state conversion at small-scale
• Catalysis of entanglement and other quantum resources
• Entropy and reversible catalysis
• Fundamental Limits on Correlated Catalytic State Transformations
• Optimal Extensions of Resource Measures and their Applications
• Constraints on magic state protocols from the statistical mechanics of Wigner negativity
• Correlations in typicality and an affirmative solution to the exact catalytic entropy conjecture
• Spins-based Quantum Otto Engines and Majorisation
• Catalysis in Action via Elementary Thermal Operations
• Quantum Foundations of Classical Reversible Computing
• No-go theorem for entanglement distillation using catalysis
• Covariant catalysis requires correlations and good quantum reference frames degrade little
• Catalysis always degrades external quantum correlations
• Evaluating the Advantage of Adaptive Strategies for Quantum Channel Distinguishability
• Catalytic transformations for thermal operations
• Transcendental properties of entropy-constrained sets
• Sufficiency of Rényi divergences
• Transcendental properties of entropy-constrained sets: Part II
• Thermodynamic state convertibility is determined by qubit cooling and heating
• Catalytic transformations with finite-size environments: applications to cooling and thermometry
• Unlocking inaccessible performance of the quantum refrigerator with catalysts