Q1. How does Constructor Theory differ from the framework of Resource Theories?
Constructor theory and the resource theories framework differ in motivations andcontent. A resource theory expresses the content (or part of the content) of existing theories, such as thermodynamics, classical or quantum computation, etc., in terms of three kinds of thing: (i) the free operations – that is, operations which can be used without limitations; (ii) the resources, i.e. the remaining operations, considered expensive; and (iii) the free states, or non-resources.
The aim is to extract, and conveniently to systematise, implications of the theory to address operational issues such as: According to that theory, which resources can be converted into which others by the free operations? What are the ways in which a given conversion can be accomplished? Can these conversions happen under various constraints, e.g.: deterministically, stochastically or when a catalyst (i.e. a resource which is not consumed in the process) is present? What are suitable measures of the quality of a resource? For instance, the resource theory of asymmetry in quantum theory (where the resources are quantum states and operations violating the symmetry) can categorise constraints arising from symmetries, such as selection rules in atomic physics, Noether’s theorem, etc.
A general, category-theory based formalism for resource theories has recently been proposed encompassing the existing resource theories (of athermality, of asymmetry, ofnon-uniformity, etc. 1 ). This formalism could be called ‘resource theory’ itself, as distinct from theories of particular resources under particular existing theories.
Thus resource theory, and resource theories, derive their entire content about the physical world from the existing theories that they are resource theories of. They have no pretentions (as Lovelace would say) to originate anything. They have neither eyes to see nor tongue to speak (as Lenthall would say) except as those theories are pleased to direct them.
Constructor theory does. It not only intends to reformulate existing theories (in terms of possible/impossible tasks only); it contradicts some existing theories (such as non-local variants of quantum theory) and also makes assertions about the physical world inaddition to those implied by existing theories. For example, the constructor-theoretic expression of the First Law of Thermodynamics implies that the energy of a constructor is bounded below and above. Also, constructor theory demands a local, deterministicstructure in physical theories – while resource theory is agnostic about this issue. Indeed, while in resource theories there are many notions of ‘performable transformation’ (including stochastic ones) in constructor theory a task being possible task can only mean that a constructor for it can be built, with arbitrary accuracy and reliability.
Thus, while the resource theory of athermality takes notions such as thermal states, heat baths, temperature and equilibrium as they are defined in thermodynamics, theconstructor theory of thermodynamics (forthcoming), aims at explaining what ‘temperature’, ‘thermal states’, ‘work’, etc. mean physically, in terms of regularities of the underlying laws, expressed exclusively via possible/impossible tasks; and at improving on the existing formulations of thermodynamics, making them exact, and not approximate.
Similarly, a quantum resource theory takes as given the entities such as ‘observables’, ‘distinguishability’, ‘information’, etc, as they are defined in quantum theory. Theconstructor theory of information instead is about formulating what tasks the laws of physics must permit or forbid (i.e., make possible or impossible) for those entities to exist; and it contains new principles about those that do not follow from quantum theory, but are conjectured new laws. They precisely describe entities and processes in nature (such as information and computation, measurement and distinguishability) that in the prevailing conception can only be understood as emergent phenomena, or in vague, informal ways. Fundamental theories formulated in the prevailing conception do not (and cannot) refer to those entities at all; nor can reformulations of them, such as resource theories – no matter how general they are. Constructor theory accommodates those entities exactly, and explains them.
1. A mathematical theory of resources”, B. Coecke, T. Fritz, and R. Spekkens, http://arxiv.org/abs/1409.5531)