We propose a novel form of classification of multipartite states, in terms of the maximum degree of non-locality they can exhibit under any choice of local observables. This uses the hierarchy of notions previously introduced by Abramsky and Brandenburger: strong contextuality, logical contextuality, and probabilistic contextuality. We study n-qubit pure states. We conjecture that for more than 2 parties, all entangled states are logically contextual. We prove a number of results in support of this conjecture: (1) We show that all permutation-symmetric states are logically non-local. (2) We study the class of balanced states with functional dependencies. These states are described by Boolean functions and have a rich structure, allowing a detailed analysis, which again confirms the conjecture in this case.
We propose a novel form of classification of multipartite states, in terms of the maximum degree of non-locality they can exhibit under any choice of local observables. This uses the hierarchy of notions previously introduced by Abramsky and Brandenburger: strong contextuality, logical contextuality, and probabilistic contextuality. We study n-qubit pure states. We conjecture that for more than 2 parties, all entangled states are logically contextual. We prove a number of results in support of this conjecture: (1) We show that all permutation-symmetric states are logically non-local. (2) We study the class of balanced states with functional dependencies. These states are described by Boolean functions and have a rich structure, allowing a detailed analysis, which again confirms the conjecture in this case.
As with entanglement, different forms of Bell nonlocality arise in the multipartite scenario. These can be defined in terms of relaxations of the causal assumptions in local hidden-variable theories. However, a characterisation of all the forms of multipartite nonlocality has until now been out of reach, mainly due to the complexity of generic multipartite causal models. Here, we employ the formalism of Bayesian networks to reveal connections among different causal structures that make a both practical and physically meaningful classification possible. Our framework holds for arbitrarily many parties. We apply it to study the tripartite scenario in detail, where we fully characterize all the nonlocality classes. Remarkably, we identify new highly nonlocal causal structures that cannot reproduce all quantum correlations. This shows, to our knowledge, the strongest form of quantum multipartite nonlocality known to date. Finally, as a by-product result, we derive a non-trivial Bell-type inequality with no quantum violation. Our findings constitute a significant step forward in the understanding of multipartite Bell nonlocality and open several venues for future research.
Recently, Yu and Oh [Phys. Rev. Lett. 108, 030402 (2012)] have conjectured that the simplest set of vectors needed to prove state-independent contextuality on a qutrit requires 13 vectors. Here we first prove that a necessary and sufficient condition for a set of vectors in any dimension d≥3 to prove contextuality for a system prepared in a maximally mixed state is that the graph GC in which vertices represent vectors and edges link orthogonal ones has chromatic number χ(GG) larger than d. Then, we prove Yu and Oh's conjecture. Finally, we prove that any set satisfying χ(GG)>d assisted with d-party maximum entanglement generates nonlocality which cannot be improved without violating the no-signalling principle. This shows that any set satisfying χ(GG)>d is a valuable resource for quantum information processing.
Nonlocality is a quintessential signature of nonclassical behaviour and a resource for quantum advantages in communication and computation.The paradoxical correlations witnessed by strong nonlocality undergird the standard probabilistic form of nonlocality and provide optimal advantages in numerous informational tasks.Three-qubit systems are the simplest ones that admit strong nonlocality.Abramsky et al. (TQC, 2017) established the existence of an infinite family of three-qubit paradoxes, beyond the well-known GHZ paradox, which exhibited a novel conditional structure.In this work, we introduce several new infinite families of three-qubit paradoxes and articulate a detailed roadmap towards the complete classification of all three-qubit nonlocality paradoxes.In particular, we prove that our paradoxes exhaust all those satisfying reasonable regularity conditions.We give an example of a highly exotic paradox and place constraints on the search for new exotic paradoxes.We conjecture that all paradoxes must involve states from a one-parameter family and provide significant evidence in support of this conjecture.
In the past century, experimental discoveries have witnessed phenomena in Nature which challenge our everyday classical intuition. In order to explain these facts, quantum theory was developed, which so far has been able to reproduce the observed results. However, I believe that our understanding of quantum mechanics can be significantly improved by the search for an operational meaning behind its mathematical formulation, which would help to identify the limitations and possibilities of the theory for information processing. An intriguing property of quantum theory is its intrinsic randomness. Indeed, Einstein, Podolsky and Rosen in 1935 questioned the completeness of quantum theory. They argued the possibility of the existence of a complete theory where variables to which we have not access determine the behaviour of physical systems, and the randomness observed in quantum mechanics is then due to our ignorance of these variables. These hidden variables theories, however, were proved not to be enough for explaining the predictions of quantum theory, as shown in the no-go theorems by Bell on quantum-nonlocality and by Kochen and Specker on quantum-contextuality. In the past decades, many experiments have corroborated the nonlocal and contextual character of Nature. However, no intuition behind these phenomena has been found, in particular about what limits their strength. In fact, special relativity alone would allow for phenomena which are more nonlocal than what quantum theory allows. Hence, much effort has been devoted to find the physical properties of quantum theory that restricts these phenomena. In this thesis, we study the constraints that arise on nonlocal and contextual phenomena when a certain exclusiveness structure compatible with quantum theory is imposed in the space of events. Here, an event denotes the situation where an outcome is obtained given that a measurement is performed on the physical system. Regarding nonlocality, we introduce a notion of orthogonality that states that events involving different outcomes of the same local measurement are exclusive, and construct constraints that the correlations among observers should satisfy. We denote this by Local Orthogonality principle (LO), which is the first intrinsically multipartite principle for bounding quantum correlations. We prove that LO identifies the supra-quantum character of some bipartite and multipartite correlations, and gets close to the quantum boundary. When studying contextuality, the same abstract event may correspond to outcomes of different measurements, which introduces a non-trivial structure in the space of events. For its study, we develop a general formalism for contextuality scenarios in the spirit of the recent approach by Cabello, Severini and Winter. In our framework, nonlocality arises as a particular case of contextuality, which allows us to study a generalization of LO. Both in nonlocality and contextuality, we find close connections to problems in combinatorics and hence use graph-theoretical tools for studying correlations. Finally, this thesis also studies the detection of nonlocal correlations. Most results on quantum nonlocality focus on few particles' experiments, while less is known about the detection of quantum nonlocality in many-body systems. Standard many-body observables involve correlations among few particles, while there is still no multipartite Bell inequality to test nonlocality merely from these data. In this thesis, we provide the first proposal for nonlocality detection in many-body systems using two-body correlations. We construct families of Bell inequalities from two-body correlators, which can detect nonlocality for systems with large number of constituents. In addition, we prove violations by systems which are relevant in nuclear and atomic physics, and show how to test these inequalities by measuring global spin components, hence opening the problem to experimental realizations.
Non-locality and contextuality are key features of quantum mechanics that distinguish it from classical physics. We aim to develop a deeper, more structural understanding of these phenomena, underpinned by robust and elegant mathematical theory with a view to providing clarity and new perspectives on conceptual and foundational issues. A general framework for logical non-locality is introduced and used to prove that 'Hardy's paradox' is complete for logical non-locality in all (2,2,l) and (2,k,2) Bell scenarios, a consequence of which is that Bell states are the only entangled two-qubit states that are not logically non-local, and that Hardy non-locality can be witnessed with certainty in a tripartite quantum system. A number of developments of the unified sheaf-theoretic approach to non-locality and contextuality are considered, including the first application of cohomology as a tool for studying the phenomena: we find cohomological witnesses corresponding to many of the classic no-go results, and completely characterise contextuality for large families of Kochen-Specker-like models. A connection with the problem of the existence of perfect matchings in k-uniform hypergraphs is explored, leading to new results on the complexity of deciding contextuality. A refinement of the sheaf-theoretic approach is found that captures partial approximations to locality/non-contextuality and can allow Bell models to be constructed from models of more general kinds which are equivalent in terms of non-locality/contextuality. Progress is made on bringing recent results on the nature of the wavefunction within the scope of the logical and sheaf-theoretic methods. Computational tools are developed for quantifying contextuality and finding generalised Bell inequalities for any measurement scenario which complement the research programme. This also leads to a proof that local ontological models with `negative probabilities' generate the no-signalling polytopes for all Bell scenarios.
In this thesis we use the language of sheaf theory in order to develop a deeper understanding of some of the fundamental differences - such as entanglement, contextuality and non-locality - between quantum and classical physics. We first present, based on the work of Abramsky and Brandenburger, how sheaves, defined over certain posets of physically meaningful contexts, give a natural setting for capturing and analysing important quantum mechanical phenomena, such as quantum non-locality and contextuality. We also describe how this setting naturally leads to a three level hierarchy of quantum contextuality: weak contextuality, logical non-locality and strong contextuality. We use these insights in order to classify certain multipartite entangled states. Almost all of these turn out to be at least logically non-local, and a number of them even turn out to be strongly contextual. We further extend this result by showing that all n-qubit entangled states, with the exception of tensor products of single-qubit and bipartite maximally-entangled states, are logically non-local. Our proof is constructive: given any n-qubit state, we present an algorithm which produces n+2 local observables witnessing its logical non-locality. In the second half of the thesis we use the same basic principle of sheaves defined over physically meaningful contexts, in order to present an elegant mathematical language, known under the name of the Topos Approach, in which many quantum mechanical concepts, such as states, observables, and propositions about these, can be expressed. We then show that the language of the Topos Approach is as least as expressive, in logical terms, as traditional quantum logic. Finally, starting from a topos-theoretic perspective, we develop the construction of contextual entropy in order to give a unified treatment of classical and quantum notions of information theoretic entropy.