Christopher Perry

Abstract We study quantum dichotomies and the resource theory of asymmetric distinguishability using a generalization of Strassen's theorem on preordered semirings. We find that an asymptotic variant of relative submajorization, defined on unnormalized dichotomies, is characterized by real-valued monotones that are multiplicative under the tensor product and additive under the direct sum. These strong constraints allow us to classify and explicitly describe all such monotones, leading to a rate formula expressed as an optimization involving sandwiched Renyi divergences. As an application we give a new derivation of the strong converse error exponent in quantum hypothesis testing.

If the second law of thermodynamics forbids a transition from one state to another, then it is still possible to make the transition happen by using a sufficient amount of work. But if we do not have access to this amount of work, can the transition happen probabilistically? In the thermodynamic limit, this probability tends to zero, but here we find that for finite-sized systems, it can be finite. We compute the maximum probability of a transition or a thermodynamical fluctuation from any initial state to any final state, and show that this maximum can be achieved for any final state which is block-diagonal in the energy eigenbasis. As a bi-product, we introduce a finite set of thermodynamical monotones related to the thermo-majorization criteria which governs state transitions, and compute the work of transition in terms of them. We also find upper and lower bounds on this transition probability, in terms of the work of transition. These results, which allow one to not take the thermodynamic limit, are made possible by using and extending techniques from quantum information theory, developed in the context of understanding entanglement theory. As a consequence, our results find application in entanglement theory, and we find the amount of entanglement required (or gained) when transforming one pure entangled state into any other.