Quantifying resource in catalytic resource theory

Abstract

We consider a general resource theory that allows the use of free resource as a catalyst. We show that the amount of `resource' contained in a given state, in the asymptotic scenario, is equal to the regularized relative entropy of resource of that state, which then yields a straightforward operational meaning to this quantity. Such an answer has been long sought for in any resource theory since the usefulness of a state in information-processing tasks is directly related to the amount of resource the state possesses in the beginning. While we need to place a few assumptions in our resource theoretical framework, it is still general enough and includes quantum resource theory of entanglement, coherence, asymmetry, non-uniformity, purity, contextuality, stabilizer computation and the classical resource theory of randomness extraction as special cases. Since our resource theoretic framework includes entanglement theory, our result also implies that the amount of noise one has to inject locally in order to erase all entanglement contained in an entangled state is equal to the regularized relative entropy of entanglement, resolving an open question posted in [Groisman et al., Phys. Rev. A. 72: 032317, 2005]. On the way to prove the main result, we also quantify the amount of resource contained in a state in the one-shot setting (where one only has a single copy of the state), in terms of the smooth max-relative entropy. Our one-shot result employs a recently developed technique of convex-split lemma.

arXiv comment21 pages, version 2. Added references and made further elaborations

References in corpus (33)

• Quantum entanglement
• Quantifying Coherence
• Quantifying Entanglement
• Quantum Coherence as a Resource
• Operational Resource Theory of Coherence
• Fundamental limitations for quantum and nano thermodynamics
• The second laws of quantum thermodynamics
• Distance measures to compare real and ideal quantum processes
• The Resource Theory of Quantum States Out of Thermal Equilibrium
• On the quantum, classical and total amount of correlations in a quantum state
• The Resource Theory of Stabilizer Computation
• Min- and Max- Relative Entropies and a New Entanglement Monotone
• Correcting Quantum Errors with Entanglement
• The resource theory of informational nonequilibrium in thermodynamics
• The general structure of quantum resource theories
• Entanglement-assisted local manipulation of pure quantum states
• Low-temperature thermodynamics with quantum coherence
• A Hierarchy of Information Quantities for Finite Block Length Analysis of Quantum Tasks
• Relating the Resource Theories of Entanglement and Quantum Coherence
• Reversible transformations from pure to mixed states, and the unique measure of information
• A Resource Framework for Quantum Shannon Theory
• Entanglement Theory and the Second Law of Thermodynamics
• Second-order asymptotics for quantum hypothesis testing
• The Minimal Work Cost of Information Processing
• The asymptotic relative entropy of entanglement
• Quantum message compression with applications
• Quantum resource theories in the single-shot regime
• The apex of the family tree of protocols: Optimal rates and resource inequalities
• Catalytic Decoupling of Quantum Information
• Axiomatic approach to contextuality and nonlocality
• Disentanglement Cost of Quantum States
• Asymptotic State Discrimination and a Strict Hierarchy in Distinguishability Norms
• Symmetrizing Cost of Quantum States

Cited by in corpus (53)

• Quantum Resource Theories
• General Resource Theories in Quantum Mechanics and Beyond: Operational Characterization via Discrimination Tasks
• Operational Advantage of Quantum Resources in Subchannel Discrimination
• One-Shot Operational Quantum Resource Theory
• How to quantify a dynamical quantum resource
• Application of the Resource Theory of Channels to Communication Scenarios
• Benchmarking one-shot distillation in general quantum resource theories
• Catalytic quantum teleportation and beyond
• A chain rule for the quantum relative entropy
• Skew informations from an operational view via resource theory of asymmetry
• Quantifying the resource content of quantum channels: An operational approach
• Catalysis of entanglement and other quantum resources
• Correlation in Catalysts Enables Arbitrary Manipulation of Quantum Coherence
• Operational quantification of continuous-variable quantum resources
• Framework for resource quantification in infinite-dimensional general probabilistic theories
• The first law of general quantum resource theories
• One-Shot Manipulation of Dynamical Quantum Resources
• Resource Preservability
• General Quantum Resource Theories: Distillation, Formation and Consistent Resource Measures
• Disentanglement Cost of Quantum States
• $\alpha$-Logarithmic negativity
• Convex-split and hypothesis testing approach to one-shot quantum measurement compression and randomness extraction
• Universal limitations on implementing resourceful unitary evolutions
• Communication, Dynamical Resource Theory, and Thermodynamics
• One-shot Coherence Distillation with Catalysts
• Amplifying asymmetry with correlated catalysts
• One-Shot Yield-Cost Relations in General Quantum Resource Theories
• Every quantum helps: Operational advantage of quantum resources beyond convexity
• Arbitrary Amplification of Quantum Coherence in Asymptotic and Catalytic Transformation
• Virtual quantum resource distillation: General framework and applications
• Efficient methods for one-shot quantum communication
• Robustness- and weight-based resource measures without convexity restriction: Multicopy witness and operational advantage in static and dynamical quantum resource theories
• Catalysis in Action via Elementary Thermal Operations
• Asymptotically Consistent Measures of General Quantum Resources: Discord, Non-Markovianity, and Non-Gaussianity
• Mutual Information is not a Reliable Measure for Variations in Total Correlations
• Quantum Bell Nonlocality is Entanglement
• Resource Marginal Problems
• Hierarchy of quantum operations in manipulating coherence and entanglement
• Reliability Function of Quantum Information Decoupling via the Sandwiched Rényi Divergence
• Teleportation with Embezzling Catalysts
• Bounding the resources for thermalizing many-body localized systems
• A Compendious Review of Majorization-Based Resource Theories: Quantum Information and Quantum Thermodynamics
• Quantum process discrimination with restricted strategies
• Projective robustness for quantum channels and measurements and their operational significance
• Unified framework for continuity of sandwiched Rényi divergences
• Operational Resource Theory of Non-Markovianity
• Quantum thermodynamics with coherence: Covariant Gibbs-preserving operation is characterized by the free energy
• Communication with Quantum Catalysts
• Many-body quantum catalysts for transforming between phases of matter
• Catalytic transformations with finite-size environments: applications to cooling and thermometry
• Unlocking inaccessible performance of the quantum refrigerator with catalysts
• All non-Gaussian states are advantageous for channel discrimination: Robustness of non-convex continuous variable quantum resources
• The entropy of quantum causal networks