Bell non-locality is a powerful framework to distinguish classical, quantum and post-quantum resources, which relies on non-communicating players. Under which restriction can we have the same separations, if we allow for communication? Non-signalling state assemblages, and the fact that they can always be simultaneously purified, turned out to be the key element to restrict the simplest bipartite communication scenario, the prepare-and-measure, to the standard bipartite Bell scenario. Yet, many distinctive features of quantum theory are genuinely multipartite and cannot be reduced to two-party behaviour. In this work we are interested in extending this simultaneous purification inspired result to all multipartite communication schemes. As a first step, we unify and extend the simultaneous purification result from states to instruments and super-instruments, which are composable structures, and open up the possibility to explore more complex communication scenarios. Our main contribution is to establish that arbitrary compositions of non-signalling assemblages cannot escape the standard spatial quantum Bell correlations set. As a consequence, any interactive quantum realization of correlations outside of this set must involve at least one signalling assemblage of quantum operations, even when the resulting correlations are non-signalling.
Bell non-locality is a powerful framework to distinguish classical, quantum and post-quantum resources, which relies on non-communicating players. Under which restriction can we have the same separations, if we allow for communication? Non-signalling state assemblages, and the fact that they can always be simultaneously purified, turned out to be the key element to restrict the simplest bipartite communication scenario, the prepare-and-measure, to the standard bipartite Bell scenario. Yet, many distinctive features of quantum theory are genuinely multipartite and cannot be reduced to two-party behaviour. In this work we are interested in extending this simultaneous purification inspired result to all multipartite communication schemes. As a first step, we unify and extend the simultaneous purification result from states to instruments and super-instruments, which are composable structures, and open up the possibility to explore more complex communication scenarios. Our main contribution is to establish that arbitrary compositions of non-signalling assemblages cannot escape the standard spatial quantum Bell correlations set. As a consequence, any interactive quantum realization of correlations outside of this set must involve at least one signalling assemblage of quantum operations, even when the resulting correlations are non-signalling.
Historically, quantum mechanics has relied on the assumption of non-communicating parties, enforced via space-like separation, to witness the gap between classical and quantum descriptions. This thesis explores the boundaries of the standard Bell non-local scenario, investigating how to preserve this gap when communication is allowed. In particular, this work focuses on restricted communication, multipartite settings, and operator-algebraic formulations. Key contributions of this work include: • Dimensional Constraints: Establishing mathematical maps from standard Bell scenarios to prepare-and-measure scenarios with dimensionally bounded channels, proving that quantum bounds and self-testing properties are preserved. • Cryptographic Constraints: Analyzing compiled non-local games using quantum homomorphic encryption. The research proves asymptotic quantum soundness for all multipartite compiled games and introduces a quantitative hierarchy, similar to the NPA hierarchy, to characterize sequential correlations. • Operational Equivalences: Demonstrating that correlations arising from arbitrary compositions of non-signalling assemblages in multipartite communication schemes always reduce to standard spatial quantum Bell correlations. • Operator-Algebraic Tools: Introducing a novel chain rule for the Radon-Nikodym derivative to dilate chains of completely-positive maps, alongside the construction of a universal C*-algebra for sequential measurements. Overall, the manuscript unifies concepts from quantum foundations, complexity theory, and operator algebras to provide a rigorous mathematical framework for characterizing and certifying multipartite sequential quantum correlations.
Abstract Fundamental investigations in non-locality have shown that while the no-signaling principle alone is not sufficient to single out the set of quantum non-local correlations, local quantum mechanics and no-signaling together exactly reproduce the set of quantum correlations in the two-party Bell scenario. Here, we introduce and study an intermediate hybrid no-signaling quantum set of non-local correlations that we term HNSQ in the multi-party Bell scenario where some subsystems are locally quantum while the remaining subsystems are only constrained by the no-signaling principle. Specifically, the set HNSQ is a super-quantum set of correlations derived from no-signaling assemblages by performing quantum measurements on the trusted subsystems. We show that in contrast to the set NS of no-signaling behaviors, there exist extreme points of HNSQ in the tripartite Bell scenario that admit quantum realization. As a tool for optimization over the set HNSQ, we introduce an outer hierarchy of semi-definite programming approximations to the set following an approach put forward by Doherty–Parrilo–Spedalieri. We perform an extensive numerical analysis of the maximal violation of the facet Bell inequalities in the three-party binary input–output scenario and study the corresponding self-testing properties. In contrast to the usual no-signaling correlations, the new set allows for simple security proofs of (one-sided)-device-independent applications against super-quantum adversaries.
The study of stronger-than-quantum phenomena (i.e., postquantum) has enabled a deeper understanding of the scope of quantum theory. Much is known about the case of correlations in Bell scenarios, where the device-independent framework allowed us to explore its possibilities independently of the formalism of quantum theory. However, less is known about the phenomenon of Einstein-Podolsky-Rosen steering. Here, the `characterised parties' are assumed to describe their systems locally through the quantum formalism, which inconveniences a theory-independent description. In addition, a theorem by Gisin and Hughston, Josza and Wootters further hindered the discovery of the phenomenon. The study of postquantum steering, initiated about a decade ago, has been quite fruitful, including: the development of mathematical formalisms that frame the effect, resource theories that quantify it as a resource, and activation protocols that relate it to Bell correlations. However, all these results have a limitation in common: they apply to scenarios with only one quantum party.
Here we articulate the concept of postquantum steering for scenarios with multiple quantum parties, bringing in the missing piece to the puzzle. We provide an algorithm to certify postquantumness, which in some cases also certifies quantumness. We also define a hierarchy of semidefinite programs that bounds the set of quantum assemblages from the outside. Moreover, we show that the study of postquantum steering is fundamentally relevant since it is not just a mere mathematical curiosity allowed by the no-signalling principle, but it may arise within compositional theories beyond quantum theory. Our work further discovers a peculiarity of steering: its theory-independent description fundamentally prevents a direct connection with Bell nonlocality -- e.g., nonclassical Bell correlations do not imply nonclassical steering.
The main objective of this thesis is to explore spatial superposition as a novel resource for quantum communication. In classical physics, at any instant of time, information flows in one direction only, i.e. an information carrier transmits the message from the sender to the receiver, but not vice versa. In contrast, quantum physics opens up the possibility of putting information carriers in spatial superposition which in turn enables simultaneous bi-directional transmission of information between the sender and the receiver. Our aim is explore this possibility to arrive at new communication tasks that cannot be accomplished within classical physics, thus quantifying the use of quantum superposition as a resource for communication. More precisely, we aim at classifying simple communication protocols (e.g. involving the exchange of information between several players, with restrictions on the available resources) by means of Bell-like inequalities, which set precise limits to classical communication. On the other hand, the classification of these inequalities provides a natural operational framework to show that quantum information carriers in spatial superposition are a genuine resource for non-classical communication (i.e. by explicitly violating the bounds imposed by Bell-like inequalities).
The violations of Bell inequalities by measurements on quantum states give rise to the phenomenon of quantum non-locality and express the advantage of using quantum resources over classical ones for certain information-theoretic tasks. The relative degree of quantum violations has been well studied in the bipartite scenario and in the multipartite scenario with respect to fully local behaviours. However, the multipartite setting entails a more complex classification in which different notions on non-locality can be established. In particular, genuine multipartite non-local distributions apprehend truly multipartite effects, given that these behaviours cannot be reproduced by bilocal models that allow correlations among strict subsets of the parties beyond a local common cause. We show here that, while in the so-called correlation scenario the relative violation of bilocal Bell inequalities by quantum resources is bounded, i.e. it does not grow arbitrarily with the number of inputs, it turns out to be unbounded in the general case. We identify Bell functionals that take the form of non-local games for which the ratio of the quantum and bilocal values grows unboundedly as a function of the number of inputs and outputs.
The no-signalling principle lies at the heart of Bell-type experiments involving spacelike-separated parties. Although not all no-signalling correlations can be realized within quantum theory, the no-sinalling principle itself remains fundamental even for post-quantum theories, since its violation would imply superluminal signalling. In the spirit of device independent nature of the Bell inequalities, there has recently been growing interest in causal inequalities, which identify correlations admitting no causal explanation. Such correlations have been rigorously studied within the process-matrix formalism, which is built upon a constraint analogous to spacelike separation in Bell scenarios. We refer to this constraint as the single-interaction constraint, whereby each party interacts with the environment only once. Despite considerable effort, no fundamental guiding principle is currently known that characterizes the restrictions this constraint imposes on the correlations generated by Process Matrices. In this work, we address this gap by proposing a generalized causality principle for the bipartite case. We show that for certain cases, the generalised causality principle defines a strict subset of the set of all signalling correlations. This principle thus provides a fundamental limit on quantum and even post-quantum theories subject to the single-interaction constraint. By analogy with no-signalling, whose violation excludes certain causal structures among the parties, a violation of the generalised causality principle rules out a corresponding class of causal structures(causal structures with the single interaction constraint). More broadly, our techniques provide a general framework for deriving generalised causal constraints, applicable to a wide class of indefinite-causal-order scenarios.