The properties of thermodynamical operations

Abstract

The resource theory approach is a recently developed framework used in the study of thermodynamics for finite sized quantum systems. A simple way to describe how a quantum system evolves under energy-preserving unitaries while interacting with thermal reservoirs, has been formulated in the framework of thermal operations. Despite the conceptual simplicity of these operations, much is still unknown about their mathematical structures. In this work, we studied these mathematical structures, in an attempt to better understand how these operations, and some variants, describe quantum thermodynamics. One of the interesting phenomena that occurs in quantum thermodynamics, is something that we call super-activation. In this phenomenon, we make multiple forbidden transitions possible by combining them together. This phenomenon could, for example, be used to extract more work from two systems, while it is impossible to extract any work from the individual systems. We found conditions for when this phenomenon can occur. As a result of this, we found that qubits with trivial Hamiltonians cannot super-activate each other. On the other hand, we found a way to construct infinitely many examples of super-activation. We also investigated what happens if we combine multiples of the same forbidden transitions together. We found necessary conditions which each have to satisfy in order for this joint transition to be possible. In particular, these conditions show that this special case of super-activation can only occur in one direction. Another topic that we studied in this work is smoothed Rényi divergences. In an attempt to give an operational meaning to these quantities, we studied two special states that allow us to clarify the relation between the smoothed Rényi divergences with the possibility of a transition. These special states are the steepest state and the attest state. For a given state p, the steepest state is a state that is e-close to p in terms of trace distance and can be transformed, in the presence of a thermal bath and while conserving the total energy, to any other state that is also e-close to p. The attest state is a state that is also e-close to p in terms of trace distance, but all e-close states of p can be transformed to it. We found a way to construct this steepest state for limited values of e, and the attest state for any e.