A mathematical theory of resources

Abstract

In many different fields of science, it is useful to characterize physical states and processes as resources. Chemistry, thermodynamics, Shannon's theory of communication channels, and the theory of quantum entanglement are prominent examples. Questions addressed by a theory of resources include: Which resources can be converted into which other ones? What is the rate at which arbitrarily many copies of one resource can be converted into arbitrarily many copies of another? Can a catalyst help in making an impossible transformation possible? How does one quantify the resource? Here, we propose a general mathematical definition of what constitutes a resource theory. We prove some general theorems about how resource theories can be constructed from theories of processes wherein there is a special class of processes that are implementable at no cost and which define the means by which the costly states and processes can be interconverted one to another. We outline how various existing resource theories fit into our framework. Our abstract characterization of resource theories is a first step in a larger project of identifying universal features and principles of resource theories. In this vein, we identify a few general results concerning resource convertibility.

32 pages, many figures. v2 and v3: minor revisions

References in corpus (32)

• Quantum entanglement
• Concentrating Partial Entanglement by Local Operations
• Conditions for a class of entanglement transformations
• Fundamental limitations for quantum and nano thermodynamics
• The second laws of quantum thermodynamics
• The Resource Theory of Quantum States Out of Thermal Equilibrium
• Informational derivation of Quantum Theory
• The resource theory of quantum reference frames: manipulations and monotones
• Work extraction and thermodynamics for individual quantum systems
• Quantum Circuits Architecture
• Extending Noether's theorem by quantifying the asymmetry of quantum states
• Physics, Topology, Logic and Computation: A Rosetta Stone
• The mother of all protocols: Restructuring quantum information's family tree
• Measuring the quality of a quantum reference frame: the relative entropy of frameness
• Reversible transformations from pure to mixed states, and the unique measure of information
• A Resource Framework for Quantum Shannon Theory
• The theory of manipulations of pure state asymmetry: basic tools and equivalence classes of states under symmetric operations
• Quantum Picturalism
• Modes of asymmetry: the application of harmonic analysis to symmetric quantum dynamics and quantum reference frames
• A Characterization of Entropy in Terms of Information Loss
• Are the laws of entanglement theory thermodynamical?
• The WAY theorem and the quantum resource theory of asymmetry
• The asymmetry properties of pure quantum states
• Introducing categories to the practicing physicist
• Higher Operads, Higher Categories
• Categories for the practising physicist
• The thermodynamic cost of reliability and low temperatures: Tightening Landauer's principle and the Second Law
• A classification of incomparable states
• A Bayesian Characterization of Relative Entropy
• An information-theoretic account of the Wigner-Araki-Yanase theorem
• Categories of Quantum and Classical Channels
• Categories of Quantum and Classical Channels (extended abstract)

Cited by in corpus (209)

• Quantum Coherence as a Resource
• Quantum Resource Theories
• Robustness of coherence: An operational and observable measure of quantum coherence
• Application of a resource theory for magic states to fault-tolerant quantum computing
• Measures and applications of quantum correlations
• How to quantify coherence: Distinguishing speakable and unspeakable notions
• Robustness of asymmetry and coherence of quantum states
• Enhanced communication with the assistance of indefinite causal order
• Resource theory of quantum non-Gaussianity and Wigner negativity
• Kochen-Specker Contextuality
• General Resource Theories in Quantum Mechanics and Beyond: Operational Characterization via Discrimination Tasks
• Alternative framework for quantifying coherence
• Introduction to the book "Quantum Theory: Informational Foundations and Foils"
• Quasi-quantization: classical statistical theories with an epistemic restriction
• The completeness of quantum theory for predicting measurement outcomes
• Quantum from principles
• Maximal Coherence and the Resource Theory of Purity
• Operational one-to-one mapping between coherence and entanglement measures
• Resource Destroying Maps
• Quantifying the Imaginarity of Quantum Mechanics
• Operational Resource Theory of Quantum Channels
• Quantifying Operations with an Application to Coherence
• Quantifying magic for multi-qubit operations
• Structure of the resource theory of quantum coherence
• Gaussian quantum resource theories
• One-Shot Operational Quantum Resource Theory
• Convex geometry of quantum resource quantification
• A resource theory of quantum memories and their faithful verification with minimal assumptions
• Operational resource theory of continuous-variable nonclassicality
• Beyond heat baths: Generalized resource theories for small-scale thermodynamics
• Operational relevance of resource theories of quantum measurements
• Certification and Quantification of Multilevel Quantum Coherence
• Quantifying Bell: the Resource Theory of Nonclassicality of Common-Cause Boxes
• Quantum resource theories in the single-shot regime
• Incompatibility robustness of quantum measurements: a unified framework
• A Resource Theory for Work and Heat
• Verifying the quantumness of a channel with an untrusted device
• Benchmarking one-shot distillation in general quantum resource theories
• Passivity and practical work extraction using Gaussian operations
• Resource theories of communication
• Quantum Relative Lorenz Curves
• Dynamical Resource Theory of Quantum Coherence
• The type-independent resource theory of local operations and shared randomness
• Incompatibility breaking quantum channels
• On the statistical complexity of quantum circuits
• Dynamical Entanglement
• A no-broadcasting theorem for quantum asymmetry and coherence and a trade-off relation for approximate broadcasting
• Entanglement of a bipartite channel
• Quantifying Bell non-locality with the trace distance
• Uniqueness of noncontextual models for stabilizer subtheories
• Quantifying the resource content of quantum channels: An operational approach
• Axiomatic and operational connections between the $l_1$-norm of coherence and negativity
• Correlations as a resource in quantum thermodynamics
• Coherence distillation machines are impossible in quantum thermodynamics
• Fundamental work cost of quantum processes
• Beyond heat baths II: Framework for generalized thermodynamic resource theories
• Resource speed limits: Maximal rate of resource variation
• Beyond the thermodynamic limit: finite-size corrections to state interconversion rates
• Generic bound coherence under strictly incoherent operations
• Quantifying coherence with respect to general quantum measurements
• Universal and Operational Benchmarking of Quantum Memories
• Role of Quantum Coherence in Thermodynamics
• Noncontextual wirings
• Resource theory of contextuality for arbitrary prepare-and-measure experiments
• Operational quantification of continuous-variable quantum resources
• A generalization of majorization that characterizes Shannon entropy
• Operational Resource Theory of Nonclassicality via Quantum Metrology
• Leggett-Garg macrorealism and the quantum nondisturbance conditions
• Type-independent Characterization of Spacelike Separated Resources
• Fundamental energy requirement of reversible quantum operations
• Resource Theory of Imaginarity: New Distributed Scenarios
• Quantum Horn's lemma, finite heat baths, and the third law of thermodynamics
• Fundamental limitations on photoisomerization from thermodynamic resource theories
• Quantifying Qubit Magic Resource with Gottesman-Kitaev-Preskill Encoding
• Energetics of correlations in interacting systems
• Quantum non-Gaussianity of light and atoms
• Framework for resource quantification in infinite-dimensional general probabilistic theories
• Axioms for retrodiction: achieving time-reversal symmetry with a prior
• The first law of general quantum resource theories
• Resource theories of multi-time processes: A window into quantum non-Markovianity
• One-Shot Manipulation of Dynamical Quantum Resources
• Resource Preservability
• Tight constraints on probabilistic convertibility of quantum states
• Operational axioms for diagonalizing states
• Capacity of non-Markovianity to boost the efficiency of molecular switches
• Microcanonical thermodynamics in general physical theories
• Avoiding irreversibility: engineering resonant conversions of quantum resources
• Quantifying dynamical coherence with dynamical entanglement
• Narratives of Quantum Theory in the Age of Quantum Technologies
• Detecting coherence via spectrum estimation
• Optimal Extensions of Resource Measures and their Applications
• Understanding the interplay of entanglement and nonlocality: motivating and developing a new branch of entanglement theory
• Virtual quantum resource distillation
• Flag Additivity in Quantum Resource Theories
• Asymptotic state transformations of continuous variable resources
• Resource theory of causal connection
• Postquantum common-cause channels: the resource theory of local operations and shared entanglement
• Universal limitations on implementing resourceful unitary evolutions
• Communication, Dynamical Resource Theory, and Thermodynamics
• Quantifying quantum invasiveness
• Toward physical realizations of thermodynamic resource theories
• Information-based approach towards a unified resource theory
• Coherence of operations and interferometry
• A Resource Theory of Quantum Measurements
• Coherence Depletion in Quantum Algorithms
• Quantifying EPR: the resource theory of nonclassicality of common-cause assemblages
• Monotones in General Resource Theories
• Strong and Weak Optimizations in Classical and Quantum Models of Stochastic Processes
• A comonadic view of simulation and quantum resources
• Neither Contextuality nor Nonlocality Admits Catalysts
• There is more to quantum interferometry than entanglement
• Effects of quantum resources on the statistical complexity of quantum circuits
• Epsilon-smooth measure of coherence
• Constructor Theory of Probability
• Operational restrictions in general probabilistic theories
• A linear program for testing nonclassicality and an open-source implementation
• Moderate deviation analysis of majorisation-based resource interconversion
• Fully quantum second-law--like statements from the theory of statistical comparisons
• Incompatibility as a resource for programmable quantum instruments
• Quantum superpositions of causal orders as an operational resource
• Necessary and Sufficient Conditions on Measurements of Quantum Channels
• Examining the validity of Schatten-$p$-norm-based functionals as coherence measures
• A general theory of comparison of quantum channels (and beyond)
• A review and reformulation of macroscopic realism: resolving its deficiencies using the framework of generalized probabilistic theories
• Attainability and lower semi-continuity of the relative entropy of entanglement, and variations on the theme
• One-shot manipulation of entanglement for quantum channels
• General state transitions with exact resource morphisms: a unified resource-theoretic approach
• Transforming pure and mixed states using an NMR quantum homogeniser
• Quantum Bell Nonlocality is Entanglement
• Mutual Information is not a Reliable Measure for Variations in Total Correlations
• What it takes to shun equilibration
• Simple Bounds for One-shot Pure-State Distillation in General Resource Theories
• Using a resource theoretic perspective to witness and engineer quantum generalized contextuality for prepare-and-measure scenarios
• Categories of Empirical Models
• Information and disturbance in operational probabilistic theories
• Information-theoretic foundations of thermodynamics in general probabilistic theories
• Operational locality in global theories
• Quantum versus classical transport of energy in coupled two-level systems
• Emergence of Constructor-based Irreversibility in Quantum Systems: Theory and Experiment
• Complete Characterization of Entanglement Embezzlement
• Censorship of Quantum Resources in Quantum Networks
• Compositional resource theories of coherence
• Rademacher complexity of noisy quantum circuits
• Accessible bounds for general quantum resources
• Boosting Biomolecular Switch Efficiency With Quantum Coherence
• Quantum Supermaps are Characterized by Locality
• Optimization free measures of quantum resources
• Thermodynamically-Efficient Local Computation and the Inefficiency of Quantum Memory Compression
• Coend Optics for Quantum Combs
• A resource theoretical unification of Mpemba effects: classical and quantum
• Closing Bell: Boxing black box simulations in the resource theory of contextuality
• Categorical composable cryptography
• Homotopy Theoretic and Categorical Models of Neural Information Networks
• Building Multiple Access Channels with a Single Particle
• A Foundation for Ledger Structures
• Quantum coherence between subspaces: State transformation, Cohering Power, $k$-coherence and other properties
• An introductory review on resource theories of generalized nonclassical light
• Constructor Theory of Thermodynamics
• An Operational Environment for Quantum Self-Testing
• Extremal elements of a sublattice of the majorization lattice and approximate majorization
• A Compendious Review of Majorization-Based Resource Theories: Quantum Information and Quantum Thermodynamics
• Categorical composable cryptography: extended version
• Every non-signalling channel is common-cause realizable
• Higher order singular value decomposition and the reduced density matrices of three qubits
• Covariant influences for finite discrete dynamical systems
• Quantum resource-theoretical analysis of the role of vibrational structure in photoisomerization
• A Framework for Universality in Physics, Computer Science, and Beyond
• Operational approach to the topological structure of the physical space
• The Dynamics of Entropies at the Onset of Interactions
• Coherence, superposition, and Löwdin symmetric orthogonalization
• Choi-Defined Resource Theories
• Thermodynamic state convertibility is determined by qubit cooling and heating
• Extending Resource Monotones using Kan Extensions
• Entropic partial orderings of quantum measurements
• Process-theoretic characterisation of the Hermitian adjoint
• Correlated dynamics as a resource in molecular switches
• Quantifying imaginarity of quantum operations
• Causality in Higher Order Process Theories
• Strong superadditivity relations for multiqubit systems
• Additive monotones for resource theories of parallel-combinable processes with discarding
• Exposure of subtle multipartite quantum nonlocality
• Detecting quantum resources in a semi-device independent framework
• On a 2-relative entropy
• Necessary and Sufficient Condition for the Equivalence of Two Pure Multipartite States under Stochastic Local Incoherent Operations and Classical Communications
• Concurrent Process Histories and Resource Transducers
• Characterising two-sided quantum correlations beyond entanglement via metric-adjusted f-correlations
• A Categorical View on the Principle of Relativity
• Collages of String Diagrams
• Protocol Choice and Iteration for the Free Cornering
• Contextuality, Witness of Quantum Weirdness
• Entropic and operational characterizations of dynamic quantum resources
• Thermodynamic criteria for signaling in quantum channels
• Compositional Separation of Control Flow and Data Flow
• From quantum foundations via natural language meaning to a theory of everything
• Quantum de Finetti Theorems as Categorical Limits, and Limits of State Spaces of C*-algebras
• Phase Transitions in Light-Matter Systems for Quantum Sensing
• Categories of Brègman operations and epistemic (co)monads
• On the classification of two-qubit group orbits and the use of coarse-grained 'shape' as a superselection property
• Mathematical methods for resource-based type theories
• Infinite dimensionality of the post-processing order of measurements on a general state space
• A Profunctorial Semantics for Quantum Supermaps
• Situated Transition Systems
• Functoriality of Quantum Resource Theory and Variable-Domain Modal Logic
• Morphisms in categories of nonlocal games
• Measure of set imaginarity
• Convexity of noncontextual wirings and how they order the set of correlations
• A Quantum Interpretation of Bunched Logic for Quantum Separation Logic
• Analysis of Neural Network Predictions for Entanglement Self-Catalysis
• CircuitFlow: A Domain Specific Language for Dataflow Programming (with appendices)