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A mathematical theory of resources
Bob Coecke, Tobias Fritz, Robert W. Spekkens
TL;DR
Resource theories need a general framework for describing costly resources, their free transformations, convertibility, rates, and quantification. The paper formalizes resource theories with symmetric monoidal categories, derives them from partitioned process theories, and identifies general structural results while leaving epsilonification for future work.
Problem
Existing resource theories address conversion, rates, catalysts, and quantification across diverse fields, motivating a general mathematical definition and shared framework.
Method
The paper models resources as objects and transformations as morphisms in a symmetric monoidal category, then constructs resource theories from partitioned process theories with distinguished free processes.
Results
The core theory of resource convertibility is a commutative preordered monoid, with preorder capturing conversion and monoid structure capturing parallel composition.
Takeaways & Limitations
The framework supports abstract study of resource convertibility and shows that preorder and parallel-composition structures can vary independently.
Takeaways & Limitations
The framework does not yet fully address epsilonification, where resources need only be converted to targets up to arbitrarily small error.
Abstract
from arXiv · showhide
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.
1 Introduction
The paper formalizes resource theories as mathematical structures for analyzing free processes, resource conversion, composition, and quantification. It derives resource theories from partitioned process theories and identifies general convertibility results in an abstract framework.
- Framework: The authors model a resource theory as a symmetric monoidal category whose objects are resources and morphisms are resource transformations.This formalizes parallel and sequential composition of resources and processes.
- Construction: Partitioned process theories distinguish free processes from costly ones and thereby define resource theories of states and generic processes.Generic processes include states, transformations, and measurements, with possible parallel or sequential composition.
- Convertibility: A theory of resource convertibility retains the preorder and commutative preordered monoid needed to answer core conversion questions without specifying the particular free operation used.The preorder captures convertibility, while the monoid specifies parallel composition.
- Convertibility: Resource monotones respect the convertibility preorder, but the preorder is more fundamental because incomparable resources cannot be characterized by a single valuation.Pure bipartite entanglement illustrates this distinction through the majorization preorder of reduced-state spectra.
- General results: The abstract framework supports general results about resource phenomena, including a sufficient condition for the absence of catalysis and questions about asymptotic conversion rates.The paper presents these results as initial steps toward identifying common principles across resource theories.
2 Resource theories
A resource theory models resources as objects and cost-free transformations as morphisms in a symmetric monoidal category. Sequential and parallel composition, a void resource, and free resources provide the framework for analyzing concrete theories.
- Composition: Transformations compose sequentially when types match, while tensor composition executes resources or processes in parallel.For f : A → B and g : B → C, sequential composition converts A to C; f ⊗ g represents parallel execution.
- Graphical calculus: The graphical calculus depicts resources as wires and transformations as boxes, with sequential composition by connecting wires and parallel composition by placing boxes side-by-side.Equational reasoning corresponds to deforming diagrams without changing their topology.
- Definition: A resource theory is represented by a symmetric monoidal category whose objects are resources and whose morphisms are transformations between them.The category provides the abstract structure for formalizing resource composition and conversion.
- Unit and graphical calculus: The tensor unit I denotes the void resource and satisfies A ⊗ I = A = I ⊗ A.A morphism from I to A has no input wires and represents preparation of A from nothing.
- Free resources: Free resources are exactly objects obtainable from the void resource through a cost-free morphism, namely those A for which D(I, A) is non-empty.Resources outside this set are costly or nonfree.
- Examples: The framework applies to concrete theories such as chemistry, where objects are collections of chemical species and morphisms are reactions or reaction sequences.The chemical tensor denotes combining species, and different reaction sequences correspond to different morphisms.
3 Resource theories from partitioned process theories
The paper constructs resource theories from partitioned process theories: a symmetric monoidal process theory together with a distinguished subtheory of free processes. This framework yields resource theories of states and processes, including parallel-combinable and universally-combinable variants.
- Process-theory examples: For classical stochastic processes, finite sets are objects, stochastic maps are morphisms, sequential composition is matrix multiplication, and parallel composition uses Cartesian products.The unit object is the singleton set, giving the symmetric monoidal category (FinStoch, ◦, ⊗, I).
- Partitioned process theories: A partitioned process theory is a symmetric monoidal category of processes paired with an all-object-including symmetric monoidal subcategory of free processes.The free subtheory is closed under parallel and sequential composition; processes outside it are costly resources, and nontriviality requires Cfree ≠ C.
- Resource theories of states: Theorem 3.4 shows that every partitioned process theory defines a symmetric monoidal resource theory of states.A state is a process whose input is the trivial object, or equivalently a preparation procedure.
- Resource theories of states: In the resulting examples, free states include separable bipartite states, G-invariant states, and Gibbs states at the reference temperature, while nonfree states are respectively entangled, G-noninvariant, or athermality resources.For entanglement, separable states have the form of mixtures of product states; for asymmetry, free states are invariant under the group action; for athermality, any state other than the specified Gibbs state is nonfree.
- Resource theories of processes: Theorem 3.12 shows that every partitioned process theory also defines a resource theory of processes by representing a single resource-process occurrence inside a circuit of free processes.Lemma 3.11 establishes that such circuits can be written as 1-combs with the resource process inserted into a free-process framework.
- Universally-combinable processes: The universally-combinable process theory removes restrictions on how resource processes are consumed, allowing them to be combined in arbitrary ways rather than only in parallel.This is denoted UC(C, Cfree) and is motivated by laboratory settings where equipment can be assembled in arbitrary configurations.
4 Theories of resource convertibility
The paper formalizes resource convertibility by retaining resources, their composition, and the existence of transformations while abstracting away individual processes. It then derives general results about catalysis and cloning, alongside examples showing how composition affects resource behavior.
- Definition: The resulting convertibility relation is reflexive and transitive, so direct and sequential transformations are represented by a preorder.The paper also retains parallel composition through the monoid operation on resources.
- Definition: A theory of resource convertibility records resources, a composition operation, a preorder of convertibility, and a distinguished zero resource.The preorder expresses whether one resource can be transformed into another, while the composition operation captures parallel combination.
- Definition: The construction maps a resource theory’s objects to resources and its morphisms to a binary relation indicating whether each conversion exists.This is described as a partial decategorification because it preserves only the existence of morphisms, not their identities or composition details.
- Examples: Resource theories can share the same underlying commutative monoids while differing in how composition combines quantities or qualities.The food example uses component-wise addition, whereas the proficiency example uses the order-theoretic supremum.
- Phenomenology: If a resource theory is non-interacting and quantity-like, it is catalysis-free.This provides a sufficient criterion for the absence of catalytic transformations.
- Phenomenology: In a quantity-like theory, a resource can be cloned if and only if it can be produced from nothing, so no non-trivial resource can be cloned.The condition is expressed as a ⪰a + a ⇐⇒ 0 ⪰a.
- Phenomenology: The general framework does not assume that every resource is freely disposable, because disposal itself can require costly resources.Nuclear waste is given as an example of a resource with negative value whose disposal requires treatment, storage, and decay time.
5 Quantitative concepts for theories of resource convertibility
This section develops quantitative tools for resource convertibility, including monotones, complete families of monotones, and asymptotic conversion rates. It shows how these tools detect impossible transformations, characterize convertibility, and bound achievable rates.
- Monotones: A monotone assigns numerical values to resources while preserving the convertibility ordering: if a ⪰b, then M(a) ≥M(b).Consequently, M(a) < M(b) certifies that a cannot be converted into b.
- Monotones: A family of monotones is complete when a conversion a ⪰b holds exactly when every monotone assigns at least as much value to a as to b.This addresses cases where the preorder is not totally ordered and one numerical value cannot capture all convertibility relations.
- Monotones: Every theory of resource convertibility has a complete family of monotones.The construction indexes the family by resources themselves and uses monotones that indicate whether each resource is convertible to a given target.
- Monotones: Additive monotones satisfy M(a + b) = M(a) + M(b), whereas supremal monotones satisfy M(a + b) = max{M(a), M(b)}.The paper suggests additive monotones fit quantitative theories, while supremal monotones fit quality-like theories.
- Conversion rates: Multiple-copy conversions can occur even when the corresponding single-copy conversion is impossible, producing economy-of-scale effects.The paper notes that n · a ⪰n · b may hold for some n even though k · a ⪰k · b fails for smaller k.
- Conversion rates: A conversion rate measures how many copies of a resource a are needed on average to produce one copy of b from arbitrarily many copies.The maximal rate is relevant when b is desired; if no positive-integer conversion exists, the rate is defined as 0.
- Conversion rates: Extensive monotones provide upper bounds on maximal conversion rates.The analogous result for minimal rates uses the opposite inequality direction.
6 Closing
The paper formalizes resource theories categorically and develops resource convertibility by retaining whether transformations exist. It identifies epsilonification as a major unresolved issue for approximate conversions.
- Resource theories are defined as symmetric monoidal categories whose objects are resources and morphisms transform those resources.
- Resource convertibility decategorifies categorical structure into a commutative preordered monoid capturing whether transformations of specified types exist.This framework targets questions concerning catalysis and conversion rates.
- The paper suggests that abstract resource-theory mathematics will interact strongly with concrete resource-theory phenomenology.
- Epsilonification remains an unresolved problem for applications allowing conversion to a resource b′ close to a target b, potentially with ε shrinking as copy numbers increase.The authors propose replacing the preorder with a cost function measuring closeness to the target, subject to analogous axioms.
A Proof of Theorem 3.12
The proof establishes that the process construction forms a symmetric monoidal category. It verifies composition, tensor structure, identities, symmetry, and the corresponding coherence properties using the underlying category’s graphical calculus.
- The construction’s sequential and parallel compositions are associative, with specified identity combs and tensor unit.The identity on f is (I, 1_dom(f), 1_cod(f)), and the tensor unit is 1_I.
- Bifunctoriality of the tensor product follows from bifunctoriality in the underlying category.
- The two orders of parallel and sequential composition produce equal composite combs by invariance of the graphical calculus for symmetric monoidal categories.
- The symmetry on f ⊗ g is represented by a comb built from free symmetry processes.These processes are free because the distinguished free processes form a sub-symmetric monoidal category.
- Naturality of the symmetry follows because applying symmetry before or after the comb transformation yields equal diagrams.