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Purity in microcanonical thermodynamics: a tale of three resource theories

Resource theories provide a framework for thermodynamics beyond the thermodynamic limit. For systems with definite energy,the key resource is purity. In this paper we formulate three alternative resource theories of purity in the framework of general probabilistic theories (GPTs). Each resource theory corresponds to a different choice of free operations, including i)random reversible operations, ii)noisy operations, generated by reversible interactions with ancillas in the microcanonical state, and iii)unital operations, which preserve the microcanonical state.We determine the requirements the underlying GPT must satisfy in order to support well-behaved resource theories with the above sets as free operations. Then, we focus on a special class of physical theories, called sharp theories with purification, which are appealing for an information-theoretic foundation of thermodynamics. For every sharp theory with purification we show that the convertibility of states implies the validity of a majorisation condition. Moreover, we characterise exactly when majorisation is sufficient for convertibility via random reversible operations. Under this condition, we prove that i) random reversible, noisy, and unital operations obey inclusion relations like in quantum theory, ii) despite being different, the three sets of operations are equivalent in terms of convertibility, iii) there exists a one-to-one correspondence between measures of purity and Schur-convex functions, and iv) purity resources are dual to entanglement resources.