Resource Theories as Quantale Modules

Monotones in General Resource Theories

“…On the other hand, we do not wish to obscure the concepts presented by needless abstraction. A more general framework that aims to broaden the scope of future investigations of resource theories is presented in the PhD thesis of one of the authors [16] instead. There, one can also find an extensive analysis of the connection of our framework for resource theories presented here to those of ordered commutative monoids [11,15] and partitioned process theories [11].…”

“…In spite of the attractiveness of resource theories with universal composition, a restriction on the types of resources being considered is often helpful when we aim to answer concrete questions of limited scope. For example, if we are only ever interested in transitions between states, it suffices to use a resource theory of states that may be modelled as a partitioned process theory [11,Section 3.2] or a quantale module [16,Example 3.13], the latter of which is closer to the analysis presented here. The key point is that answers to the questions that we investigate here do not hinge on the assumption of universal composition.…”

“…The key point is that answers to the questions that we investigate here do not hinge on the assumption of universal composition. Indeed, one can develop them in the more general setting of quantale modules [16] as well as other approaches. The minimal framework we use allows us to describe the results in a clear manner devoid of unnecessary structures, but the ideas can equally well be applied in the context of type-restricted resource theories such as resource theories of states.…”

“…Rather, they are instances of frameworks in which there is a conceptual distinction between the objects that constitute resources and those that constitute transformations. Such frameworks include resource theories of states arising from partitioned process theories [11] as well as resource theories defined through quantale modules [16]. Nonetheless, such concrete type-specific resource theories can in general be embedded into a universally combinable one.…”

“…This notion of embedding implies that results derived in the framework of universally combinable resource theories can be carried over to address questions in concrete typespecific resource theories. See [16,Chapter 4] for more details on these generalizations.…”