Quantum information theory is gaining interest in different communities where it can provide new tools for computation and concurrency, security, and communication protocols, as well as methods to deal with cognition and decision-making. All these domains take advantage of properties that characterise the quantum framework such as superposition, entanglement, and contextuality. A necessary step to exploit the potential advantage of quantum information theory is to highlight the resources underlying this advantage in a theory-independent way. Indeed, the sharing of quantum resources beyond the local scale takes place in current information and communication systems, which are grounded on classic resources: synergy between these two domains is relevant to prompt the diffusion of quantum methodologies in real applications. The present contribution aims to explore different domains where non-quantum and quantum resources coexist and non-classicality arises in different forms. Here, the term “non-classicality” may refer to the elaboration of information through classical or hybrid quantum-classical computing, but also to the representation of knowledge and applications to decision-making under uncertainty. The objective of this contribution is to identify common structures and formal methods that can be specified in different domains sharing the same technological framework. In this way, non-classicality can connect such domains and represent, by itself, a type of resource, supporting the definition of new techniques to deal with distributed classical and quantum information.
Quantum information theory is gaining interest in different communities where it can provide new tools for computation and concurrency, security, and communication protocols, as well as methods to deal with cognition and decision-making. All these domains take advantage of properties that characterise the quantum framework such as superposition, entanglement, and contextuality. A necessary step to exploit the potential advantage of quantum information theory is to highlight the resources underlying this advantage in a theory-independent way. Indeed, the sharing of quantum resources beyond the local scale takes place in current information and communication systems, which are grounded on classic resources: synergy between these two domains is relevant to prompt the diffusion of quantum methodologies in real applications. The present contribution aims to explore different domains where non-quantum and quantum resources coexist and non-classicality arises in different forms. Here, the term “non-classicality” may refer to the elaboration of information through classical or hybrid quantum-classical computing, but also to the representation of knowledge and applications to decision-making under uncertainty. The objective of this contribution is to identify common structures and formal methods that can be specified in different domains sharing the same technological framework. In this way, non-classicality can connect such domains and represent, by itself, a type of resource, supporting the definition of new techniques to deal with distributed classical and quantum information.
This thesis establishes a number of connections between foundational issues in quantum theory, and some quantum information applications. It starts with a review of quantum contextuality and non-locality, multipartite entanglement characterisation, and of a few quantum information protocols. Quantum non-locality and contextuality are shown to be essential for different implementations of quantum information protocols known as quantum random access codes and quantum communication complexity protocols. I derive sufficient experimental conditions for tests of these quantum properties. I also discuss how the distribution of quantum information through quantum cloning processes can be useful in quantum computing. Regarding entanglement characterisation, some results are obtained relating two problems, that of additivity of the relative entropy of entanglement, and that of identifying different types of tripartite entanglement in the asymptotic regime of manipulations of many copies of a given state. The thesis ends with a description of an information processing task in which a single qubit substitutes for an arbitrarily large amount of classical communication. This result is interpreted in different ways: as a gap between quantum and classical computation space complexity; as a bound on the amount of classical communication necessary to simulate entanglement; and as a basic result on hidden-variable theories for quantum mechanics. I also show that the advantage of quantum over classical communication can be established in a feasible experiment.
Abstract Quantum information theory is built upon the realisation that quantum resources like coherence and entanglement can be exploited for novel or enhanced ways of transmitting and manipulating information, such as quantum cryptography, teleportation, and quantum computing. We now know that there is potentially much more than entanglement behind the power of quantum information processing. There exist more general forms of non-classical correlations, stemming from fundamental principles such as the necessary disturbance induced by a local measurement, or the persistence of quantum coherence in all possible local bases. These signatures can be identified and are resilient in almost all quantum states, and have been linked to the enhanced performance of certain quantum protocols over classical ones in noisy conditions. Their presence represents, among other things, one of the most essential manifestations of quantumness in cooperative systems, from the subatomic to the macroscopic domain. In this work we give an overview of the current quest for a proper understanding and characterisation of the frontier between classical and quantum correlations (QCs) in composite states. We focus on various approaches to define and quantify general QCs, based on different yet interlinked physical perspectives, and comment on the operational significance of the ensuing measures for quantum technology tasks such as information encoding, distribution, discrimination and metrology. We then provide a broader outlook of a few applications in which quantumness beyond entanglement looks fit to play a key role.
Quantum mechanics exhibits many counter-intuitive features, which challenge our classical understanding of the world. This makes the theory simultaneously difficult to grasp, and a source of great potential for new technologies. We study various characteristics of classical and quantum mechanics and their respective roles in computation. This is interesting from a philosophical perspective, as it brings us a small step closer to understanding the nature of quantum mechanics. Furthermore, it leads us to discover scenarios where quantum resources provide a genuine advantage over classical resources, as well as more intuitive ways to model quantum protocols. Firstly, we explore the concept of quantum contextuality, which distinguishes uniquely-quantum experimental outcomes from outcomes that can be realised classically. In pursuit of a universal theory of contextuality, we unify two abstract frameworks: the sheaf-theoretic approach to contextuality by Abramsky and Brandenburger and the equivalence-based approach by Spekkens. We apply the sheaf-theoretic formalism to the field of communication complexity. We derive a bound on the success probability of a communication complexity protocol, based on the amount of contextuality present in its quantum resource. We provide a set of necessary conditions for distributed functions that allow a contextual advantage. Secondly, we consider a family of bicategorical models to describe the interaction of classical and quantum information in quantum protocols. We prove that this family is symmetric monoidal, and therefore suitable for describing compound and parallel protocols. This family serves as a mathematical foundation for the planar language for quantum protocols by Vi- cary. Important examples of this family are the bicategories 2Hilb and 2Vect. Additionally, we introduce the structure 2[CP∗[FHilb]], which contains a symmetric monoidal bicategory of mixed states and completely positive maps. This allows a unified description of quantum teleportation and classical encryption in a single 2-category, as well as a universal security proof applicable simultaneously to both scenarios. As part of the process, we develop extensive categorical machinery for constructing symmetric monoidal structures of bicategories, pseudo functors, pseudo transformations and modifications. We consider a method of constructing symmetric monoidal bicategories from symmetric monoidal double categories. We show that this method can be generalised to a functor from the locally cubical bicategory of monoidal double categories to the locally cubical bicategory of monoidal bicategories.
Despite the success of quantum mechanics in predicting the outcomes of experiments in many branches of physics, the foundations of the theory have remained subject of research and dispute. At the basis of this struggle with the theory lie the phenomena of nonlocality and entanglement. Since it was first predicted by Bell in 1964, nonlocality was not only verified in numerous experiments, but also identified as a useful resource for quantum information processing. Thus, the study of nonlocality is important both from a fundamental point of view and with respect to new applications in quantum information theory, such as secure cryptography and randomness generation. The identification of entanglement as a resource for quantum information led to a strong theoretical effort devoted to its characterisation and detection. Many of the resulting mathematical tools find application in several domains of physics. Although the only known way to create nonlocal correlations is to measure entangled quantum systems, it has been shown that entanglement and nonlocality constitute two inequivalent properties. Therefore, in the light of the success of entanglement theory, it is of interest to also devise a resource theory of nonlocality. The task of this thesis is to develop such a theoretical framework for the characterisation of nonlocality as a resource. To gain a better understanding of nonlocal correlations it will be necessary to investigate correlation scenarios that go beyond the situation originally considered by Bell. In doing so, this thesis provides new description of nonlocality that also have implications for the characterisation of quantum correlations and the detection of new forms of nonlocality. The first question we address is how nonlocality can consistently be defined in a scenario of arbitrarily many parties that may collaborate among each other. To this end we recognise which are the allowed physical operations in this situation and then define nonlocality as the resource that cannot be created by these operations. Our approach shows that the conventional definition of multipartite nonlocality, adopted by the community so far, is inconsistent with this operational definition; we further propose and analyse new models that do not suffer from these inconsistencies. Furthermore, we show that our findings have implications for the characterisation of quantum correlations. A recent approach to describe the set of quantum correlation consists in using principles inspired from information theory. By using a special instance of the models we defined earlier we show a fundamental limitation of this approach: no bipartite information principle is sufficient to single out the set of quantum correlations from the set of nonsignalling correlations. We then developed a description of nonlocality in an even more generalised scenario of several parties. Motivated by a result of Popescu we study scenarios where the parties are allowed to perform not only a single but sequences of measurements on their systems. Characterising nonlocality also in this scenario in operational terms and defining local models compatible with this definition, we show that a new form of nonlocality can be detected Lastly, we examine the problem of detecting the presence of nonlocality in a multipartite scenario when one is given only partial access to the global system. We find that one can verify that the total system must display nonlocality, even though the accessible subsystems only exhibit local correlations.
In this chapter, we introduce the reader to the field of quantum theory with emphasis on applications in cryptography and communication. Quantum devices are physical instruments that allow two or more parties to harness the quantum mechanical properties of superposition and entanglement to perform tasks, some considered to be impossible with non-quantum, or classical devices. In this chapter, we will learn the fundamental differences between quantum and classical processes and introduce some quantum protocols which demonstrate the advantages of quantum devices over their classical counterparts. After introducing the basics of quantum theory, we cover several topics of interest in quantum information. We begin with an overview of super-dense coding, quantum teleportation, and quantum key distribution, communication protocols that each demonstrate the security features of quantum communication. We then peer into the realm of current research by first introducing quantum error-correction and its connections to some old problems in graph theory (Weaver's quantum Ramsey theorem), and then introduce the topic of non-local games and quantum correlations and discuss their role in device-independent quantum cryptography. We conclude with a historical survey of the field, providing many references for the reader to more deeply explore the practical and theoretical sides of quantum information.
Quantum entanglement is the key property that makes quantum information theory different from its classical counterpart and is also a valuable physical resource with massive potential for technological applications. However, our understanding of entanglement is still far from complete despite intense research activities. Like other physical resources, the first step towards exploiting them fully is to know how to quantify. There are many reasons to focus on the entanglement of continuous-variable states since the underlying degrees of freedom of physical systems carrying quantum information are frequently continuous, rather than discrete. Much of the effort has been concentrated on Gaussian states, because these are common as the ground or \nthermal states of optical modes. Within this framework, many interesting topics have been studied and some significant progress made. Nevertheless, non-Gaussian states are also extremely important; this is especially so in condensed-phase systems, where harmonic behaviour in any \ndegree of freedom is likely to be only an approximation. So far, there is little knowledge about the quantification of entanglement in non-Gaussian states. \nThis thesis aims to contribute to the active field of research in quantum entanglement by introducing a new approach to the analysis of entanglement, especially in continuous-variable states, and shows that it leads to the first systematic quantification of the (local) entanglement \nin arbitrary bipartite non-Gaussian states. By applying this local approach, many new insights can be gained. Notably, local entanglements of systems with smooth wavefunctions are fully characterised by the derived simple expressions, provided the wavefunction is known. The local (logarithmic) negativity of any two-mode mixed states can be directly computed from the closed-form formulae given. For multi-mode mixed states, this approach provides a scheme that \npermits much simpler numerical computation for quantifying entanglement than is generally possible from directly computing the full entanglement of the system.