Quantum communication tasks

Abstract

Quantum theory is one of the most important theories in modern physics, yet the physical principles underlying the theory are anything but clear. Nonclassical features, such as entanglement, are often attributed as the key phenomena that make quantum theory special. However, the difference between classical and quantum systems can already be detected by observing the behavior of single systems in various communication setups.

This thesis is based on the original publications I–IV. A key concept throughout is that of a communication task, by which we mean a description of conditional probabilities in a prepare-and-measure scenario. These conditional probabilities are conveniently collected into row-stochastic matrices, which we call communication matrices.

The concept of a communication task was introduced in Publication II where we also studied a preorder on the set of communication matrices. We called this preorder the ultraweak matrix majorization and refined the concept in Publication III. A key motivation for introducing this preorder was that the set of communication matrices is closed with respect to the ultraweak matrix majorization. Additionally, ultraweak matrix majorization can be used to give a physical characterization of which communication tasks are harder to implement than others.

We also studied monotone functions of the ultraweak preorder. By studying the different monotones it becomes possible to define different notions of dimension for operational theories. These dimensions each characterize the properties of given operational theories and we are able to capture some key differences between classical and quantum state spaces. While the preorder of ultraweak matrix majorization is a major part of this thesis, some concrete communication tasks are also studied. One of the main studied communication tasks is antidistinguishability, which plays an important role in the study of the foundations of quantum mechanics. We were able to provide a new algebraic condition for an arbitrary set of quantum states to be antidistinguishable in Publication I. We also apply the theory of ultraweak matrix majorization to antidistinguishability in the third chapter of this thesis, where we show that the set of all communication matrices is not convex for classical or quantum state spaces in any dimension.

The other communication tasks studied in this thesis are communication of partial ignorance, studied in Publication II, and partial-ignorance communication tasks which was the topic of Publication IV. Both of these communication tasks can be seen as communication tasks between two parties, where one party is trying to communicate which choices the other party should avoid. A key observation for these tasks is that they lie between distinguishability and antidistinguishability. Some novel analysis is presented for both of these tasks in the final chapter of this thesis. The quantum implementation for one of the partial-ignorance communication tasks can be shown to break the principle of noncontextuality, thus proving that quantum mechanics holds a contextual advantage in the given task when compared to classical operational theories.