Resource convertibility and ordered commutative monoids

Abstract

Resources and their use and consumption form a central part of our life. Many branches of science and engineering are concerned with the question of which given resource objects can be converted into which target resource objects. For example, information theory studies the conversion of a noisy communication channel instance into an exchange of information. Inspired by work in quantum information theory, we develop a general mathematical toolbox for this type of question. The convertibility of resources into other ones and the possibility of combining resources is accurately captured by the mathematics of ordered commutative monoids. As an intuitive example, we consider chemistry, where chemical reaction equations such as \[ \mathrm{2H_2 + O_2} \to \mathrm{2H_2O} \] are concerned both with a convertibility relation "$\to$" and a combination operation "$+$". We study ordered commutative monoids from an algebraic and functional-analytic perspective and derive a wealth of results which should have applications to concrete resource theories, such as a formula for rates of conversion. As a running example showing that ordered commutative monoids are also of purely mathematical interest, we exemplify our results with the ordered commutative monoid of graphs. While closely related to both Girard's linear logic and to Deutsch's constructor theory, our framework also produces results very reminiscent of the utility theorem of von Neumann and Morgenstern in decision theory and of a theorem of Lieb and Yngvason on thermodynamics. Concerning pure algebra, our observation is that some pieces of algebra can be developed in a context in which equality is not necessarily symmetric, i.e. in which the equality relation is replaced by an ordering relation. For example, notions like cancellativity or torsion-freeness are still sensible and very natural concepts in our ordered setting.

63 pages. v2: revised exposition, new title

References in corpus (5)

• The resource theory of quantum reference frames: manipulations and monotones
• The general structure of quantum resource theories
• The mother of all protocols: Restructuring quantum information's family tree
• Multiple-copy entanglement transformation and entanglement catalysis
• Entropy Meters and the Entropy of Non-extensive Systems

Cited by in corpus (42)

• Quantum Resource Theories
• The role of quantum information in thermodynamics --- a topical review
• Quantifying the Imaginarity of Quantum Mechanics
• One-Shot Operational Quantum Resource Theory
• Quantifying Bell: the Resource Theory of Nonclassicality of Common-Cause Boxes
• Incompatibility robustness of quantum measurements: a unified framework
• A Resource Theory for Work and Heat
• Quantifying the resource content of quantum channels: An operational approach
• Optimized quantum f-divergences and data processing
• Resource theory of contextuality for arbitrary prepare-and-measure experiments
• Amortized entanglement of a quantum channel and approximately teleportation-simulable channels
• Entanglement and secret-key-agreement capacities of bipartite quantum interactions and read-only memory devices
• The first law of general quantum resource theories
• Microcanonical thermodynamics in general physical theories
• The asymptotic spectrum of graphs and the Shannon capacity
• Majorization uncertainty relations for mixed quantum states
• A Resource Theory of Quantum Measurements
• Monotones in General Resource Theories
• A comonadic view of simulation and quantum resources
• Neither Contextuality nor Nonlocality Admits Catalysts
• Mutual Information is not a Reliable Measure for Variations in Total Correlations
• A generalization of Strassen's Positivstellensatz
• Catalysis always degrades external quantum correlations
• Asymptotic majorization of finite probability distributions
• Abstract Vergleichsstellensätze for preordered semifields and semirings I
• Compositional resource theories of coherence
• A family of multipartite entanglement measures
• Monotone additive statistics
• A unified construction of semiring-homomorphic graph invariants
• Closing Bell: Boxing black box simulations in the resource theory of contextuality
• Categorical composable cryptography
• Matrix majorization in large samples
• Relating compatibility and divisibility of quantum channels
• Smooth entropy in axiomatic thermodynamics
• Extremal elements of a sublattice of the majorization lattice and approximate majorization
• Operational approach to the topological structure of the physical space
• Sufficiency of Rényi divergences
• An algebraic theory for data linkage
• Detecting quantum resources in a semi-device independent framework
• Convexity of noncontextual wirings and how they order the set of correlations
• Analysis of Neural Network Predictions for Entanglement Self-Catalysis
• A finite sufficient set of conditions for catalytic majorization