Indefinite causal structures and Shannon theory

Abstract

At the most fundamental level, Nature obeys the laws of quantum mechanics. These laws emerged through a series of discoveries by Planck, Einstein and Bohr at the beginning of the 20th century, and was later formalized in the works by Dirac and von Neumann. On the other hand, the framework of information theory, introduced in the seminal work by Claude Shannon, gave purely classical descriptions of information. It took a quarter of a century to develop a model in which the carriers of information were quanta, laying the foundations to the new field of quantum information theory. As in classical information theory, the processing of data can be most generally described by a channel, i. e. a device acting on input data and producing an output. Generalising this concept, a great amount of effort is nowadays devoted to the implementation of quantum communication networks, for which information is processed by a collective of interlinked quantum channels. With the rapid progress on this front, it is natural to expect physical realisations in the near future.

This thesis presents an extensive study of quantum networks, from three different viewpoints. Firstly, a framework for a systematic optimisation of networks is introduced. It is based on semidefinite programming for which efficient solutions are known. The framework holds for arbitrary linear figures of merit and is applicable for both causal networks, for which the devices operate in a fixed order, and non-causal networks, for which the order is indefinite. Besides, optimal figures of merit are connected to entropic measures, yielding a generalisation of the max relative entropy, originally introduced by Renner and Datta for quantum states, to networks. Building on this connection, an extension of the conditional min-entropy, defined by the same authors, to arbitrary causal networks is defined as a quantification of correlations a causal network can generate.

Secondly, quantum Shannon theory is studied in the light of non-causal networks. So far, quantum Shannon theory considered information encoded in quantum states, while the communication paths remained classical. In this thesis, I outline the foundations of a fully quantum Shannon theory, showing that a quantisation of the paths can increase the communication rate beyond the limits of conventional quantum Shannon theory. To this purpose, it is shown that the classical, entanglement-assisted classical, quantum and private capacity of a channel can be boosted. For the latter two quantities, this has further-reaching implications in terms of enhanced security of communication.

Finally, the question of how to extract information from a network, given that it is unknown or untrusted, is investigated. Here, a basic problem is to find out in which direction the information is transferred by the network. In this thesis, the task of causal hypothesis testing is defined as an instance of causal inference. More precisely, given a set of candidates describing the direction of information flow, how to identify the correct one? It is shown that quantum mechanics can lead to an exponentially smaller probability of error in this task compared to the best classical strategies.