by Brendan Fong
Hugo Nava-Kopp and I have a new paper on resource theories:
• Brendan Fong and Hugo Nava-Kopp, Additive monotones for resource theories of parallel-combinable processes with discarding.
A mathematical theory of resources is Tobias Fritz’s current big project. He’s already explained how ordered monoids can be viewed as theories of resource convertibility in a three part series on this blog.
Ordered monoids are great, and quite familiar in network theory: for example, a Petri net can be viewed as a presentation for an ordered commutative monoid. But this work started in symmetric monoidal categories, together with my (Oxford) supervisor Bob Coecke and Rob Spekkens.
The main idea is this: think of the objects of your symmetric monoidal category as resources, and your morphisms as ways to convert one resource into another. The monoidal product or ‘tensor product’ in your category allows you to talk about collections of your resources. So, for example, in the resource theory of chemical reactions, our objects are molecules like oxygen O2, hydrogen H2, and water H2O, and morphisms things like the electrolysis of water:
This is a categorification of the ordered commutative monoid of resource convertibility: we now keep track of how we convert resources into one another, instead of just whether we can convert them.
Categorically, I find the other direction easier to state: being a category, the resource theory is enriched over , while a poset is enriched over the poset of truth values or ‘booleans’ If we ‘partially decategorify’ by changing the base of enrichment along the functor that maps the empty set to 0 and any nonempty set to 1, we obtain the ordered monoid corresponding to the resource theory.
But the research programme didn’t start at resource theories either. The starting point was ‘partitioned process theories’.
Here’s an example that guided the definitions. Suppose we have a bunch of labs with interacting quantum systems, separated in space. With enough cooperation and funding, they can do big joint operations on their systems, like create entangled pairs between two locations. For ‘free’, however, they’re limited to classical communication between the locations, although they can do the full range of quantum operations on their local system. So you’ve got a symmetric monoidal category with objects quantum systems and morphisms quantum operations, together with a wide (all-object-including) symmetric monoidal subcategory that contains the morphisms you can do with local quantum operations and classical communication (known as LOCC operations).
This general structure: a symmetric monoidal category (or SMC for short) with a wide symmetric monoidal subcategory, is called a partitioned process theory. We call the morphisms in the SMC processes, and those in the subSMC free processes.
There are a number of methods for building a resource theory (i.e. an SMC) from a partitioned process theory. The unifying idea though, is that your new SMC has the processes as objects, and morphisms ways of using the free processes to build from
But we don’t have to go to fancy sounding quantum situations to find examples of partitioned process theories. Instead, just look at any SMC in which each object is equipped with an algebraic structure. Then the morphisms defining this structure can be taken as our ‘free’ processes.
For example, in a multigraph category every object has the structure of a ‘special commutative Frobenius algebra’. That’s a bit of a mouthful, but John defined it a while back, and examples include categories where morphisms are electrical circuits, and categories where morphisms are signal flow diagrams.
So these categories give partitioned process theories! This idea of partitioning the morphisms into ‘free’ ones and ‘costly’ ones is reminiscent of what I was saying earlier about the operad of wiring diagrams about it being useful to separate behavioural structure from interconnection structure.
This suggests that we can also view the free processes as generating some sort of operad, that describes the ways we allow ourselves to use free processes to turn processes into other processes. If we really want to roll a big machine out to play with this stuff, framed bicategories may also be interesting; Spivak is already using them to get at questions about operads. But that’s all conjecture, and a bit of a digression.
To get back to the point, this was all just to say that if you find yourself with a bunch of resistors, and you ask ‘what can I build?’, then you’re after the resource theory apparatus.
You can read more about this stuff here:
• Bob Coecke, Tobias Fritz and Rob W. Spekkens, A mathematical theory of resources.
• Tobias Fritz, The mathematical structure of theories of resource convertibility I.