Contextuality is a key feature of quantum mechanics that provides an important non-classical resource for quantum information and computation. Abramsky and Brandenburger used sheaf theory to give a general treatment of contextuality in quantum theory [New Journal of Physics 13 (2011) 113036]. However, contextual phenomena are found in other fields as well, for example database theory. In this paper, we shall develop this unified view of contextuality. We provide two main contributions: firstly, we expose a remarkable connection between contexuality and logical paradoxes; secondly, we show that an important class of contextuality arguments has a topological origin. More specifically, we show that "All-vs-Nothing" proofs of contextuality are witnessed by cohomological obstructions.
Contextuality is a key feature of quantum mechanics. We present the sheaf-theoretic approach to contextuality introduced by Abramsky and Brandenburger, and show how it covers a range of logical and physical phenomena "at the borders of paradox".
While quantum computers are expected to yield considerable advantages over classical devices, the precise features of quantum theory enabling these advantages remain unclear. Contextuality--the denial of a notion of classical physical reality--has emerged as a promising hypothesis. Magic states are quantum resources critical for practically achieving universal quantum computation. They exhibit the standard form of contextuality that is known to enable probabilistic advantages in a variety of computational and communicational tasks. Strong contextuality is an extremal form of contextuality desc…
Many seemingly paradoxical effects are known in the predictions for outcomes of measurements made on pre- and post-selected quantum systems. A class of such effects, which we call ``logical pre- and post-selection paradoxes'', bear a striking resemblance to proofs of the Bell-Kochen-Specker theorem, which suggests that they demonstrate the contextuality of quantum mechanics. Despite the apparent similarity, we show that such effects can occur in noncontextual hidden variable theories, provided measurements are allowed to disturb the values of the hidden variables.
We describe a joint cohomological framework for measurement-based quantum computation (MBQC) and the corresponding contextuality proofs. The central object in this framework is an element in the second cohomology group of the chain complex describing a given MBQC. It contains the function computed, up to gauge equivalence, and at the same time is a contextuality witness. The present cohomological description only applies to temporally flat MBQCs, and we outline an approach for extending it to the temporally ordered case.
Recent work by Abramsky and Brandenburger used sheaf theory to give a mathematical formulation of non-locality and contextuality. By adopting this viewpoint, it has been possible to define cohomological obstructions to the existence of global sections. In the present work, we illustrate new insights into different aspects of this theory. We shed light on the power of detection of the cohomological obstruction by showing that it is not a complete invariant for strong contextuality even under symmetry and connectedness restrictions on the measurement cover, disproving a previous conjecture. We g…
Many seemingly paradoxical effects are known in the predictions for outcomes of intermediate measurements made on pre- and post-selected quantum systems. Despite appearances, these effects do not demonstrate the impossibility of a noncontextual hidden variable theory, since an explanation in terms of measurement-disturbance is possible. Nonetheless, we show that for every paradoxical effect wherein all the pre- and post- selected probabilities are 0 or 1 and the pre- and post-selected states are nonorthogonal, there is an associated proof of contextuality. This proof is obtained by considering…
The sheaf theoretic description of non-locality and contextuality by Abramsky and Brandenburger sets the ground for a topological study of these peculiar features of quantum mechanics. This viewpoint has been recently developed thanks to sheaf cohomology, which provides a sufficient condition for contextuality of empirical models in quantum mechanics and beyond. Subsequently, a number of studies proposed methods to detect contextuality based on different cohomology theories. However, none of these cohomological descriptions succeeds in giving a full invariant for contextuality applicable to co…
This short note is an introduction to the slides from our talks given in 2018 during the Advanced School of Quantum Foundations and Quantum Computation in Joao Pessoa , Brazil. In these talks, addressed to participants with a limited knowledge of quantum foundations, we defined locality, causality, randomness, counterfactual definiteness and we explained EPR-Bohm paradoxes. In particular we discussed in great detail various Bell-type inequalities and the implications of their violation in spin polarisation correlation experiments. Finally we explained why and how the predictive completeness of…
I present a generalized notion of obstruction in Čech cohomology on semi-modules, which allows one to characterize non-disturbing contextual behaviors with any semi-field. This framework generalizes the usual Čech cohomology used in the sheaf approach to contextuality, avoiding false positives. I revise a similar work done in the framework of effect algebras with cyclic and order cohomologies and explore the relationship with the one presented here.
Contextuality and nonlocality are non-classical properties exhibited by quantum statistics whose implications profoundly impact both foundations and applications of quantum theory. In this paper we provide some insights into logical contextuality and inequality-free proofs. The former can be understood as the possibility version of contextuality, while the latter refers to proofs of quantum contextuality/nonlocality that are not based on violations of some noncontextuality (or Bell) inequality. The present work aims to build a bridge between these two concepts from what we call possibilistic p…
Okay, Roberts, Bartlett and Raussendorf recently introduced a new cohomological approach to contextuality in measurement based quantum computation. We give an abstract description of their obstruction and the algebraic structure it exploits, using the sheaf theoretic framework of Abramsky and Brandenburger. At this level of generality we contrast their approach to the Cech cohomology obstruction of Abramsky, Mansfield and Barbosa and give a direct proof that Cech cohomology is at least as powerful.
In a previous paper with Adam Brandenburger, we used sheaf theory to analyze the structure of non-locality and contextuality. Moreover, on the basis of this formulation, we showed that the phenomena of non-locality and contextuality can be characterized precisely in terms of obstructions to the existence of global sections. Our aim in the present work is to build on these results, and to use the powerful tools of sheaf cohomology to study the structure of non-locality and contextuality. We use the Cech cohomology on an abelian presheaf derived from the support of a probabilistic model, viewed …
The two state vector formalism (TSVF) was proposed by Aharonov, Bergmann, and Lebowitz (ABL) to provide a way for the counterfactual assignment of the probabilities of outcomes of contemplated but unperformed measurements on quantum systems. This formalism underlies various aspects of foundations of quantum theory and has been used significantly in the development of weak values and several proofs of quantum contextuality. We consider the application of TSVF, with pre- and post-selection (PPS) and the corresponding ABL rule, as a means to unearth quantum contextuality. We use the principle of …
We present two results on the subject of quantum contextuality and cohomology, and non-locality and quantum advantage with shallow circuits. Abramsky et al. showed that a range of examples of quantum contextuality is detected by a cohomological invariant based on Čech cohomology. However, the approach does not give a complete cohomological characterisation of contextuality. A different cohomological approach to contextuality was introduced by Okay et al. Their approach exploits the algebraic structure of the Pauli operators and their qudit generalisations known as Weyl operators. We give an ab…
Many features of single-partite quantum walks can be simulated by classical waves. However, it was recently experimentally shown that some temporal sequences of measurements on a quantum walker do not admit a classical description in terms of macro-realistic theories. Here, we extend this observation and present pre- and post-selection scenarios in quantum walks leading to logical paradoxes. Such paradoxes were recently shown to be equivalent to proofs of contextuality, therefore we provide an additional argument for non-classicality of a quantum walk model. The assumptions behind the claims o…
The data of a physical experiment can be represented as a presheaf of probability distributions. A striking feature of quantum theory is that those probability distributions obtained in quantum mechanical experiments do not always admit a joint probability distribution, a celebrated observation due to Bell. Such distributions are called contextual. Simplicial distributions are combinatorial models that extend presheaves of probability distributions by elevating sets of measurements and outcomes to spaces. Contextuality can be defined in this generalized setting. This paper introduces the notio…
The notion of contextuality, which emerges from a theorem established by Simon Kochen and Ernst Specker (1960-1967) and by John Bell (1964-1966), is certainly one of the most fundamental aspects of quantum weirdness. If it is a questioning on scholastic philosophy and a study of contrafactual logic that led Specker to his demonstration with Kochen, it was a criticism of von Neumann's "proof" that led John Bell to the result. A misinterpretation of this famous "proof" will lead them to diametrically opposite conclusions. Over the last decades, remarkable theoretical progresses have been made on…
We establish a strong link between two apparently unrelated topics: the study of conflicting information in the formal framework of valuation algebras, and the phenomena of non-locality and contextuality. In particular, we show that these peculiar features of quantum theory are mathematically equivalent to a general notion of \emph{disagreement} between information sources. This result vastly generalises previously observed connections between contextuality, relational databases, constraint satisfaction problems, and logical paradoxes, and gives further proof that contextual behaviour is not a…
A kind of paradoxical effects has been demonstrated that the pigeonhole principle, i.e., if three pigeons are put in two pigeonholes then at least two pigeons must stay in the same hole, fails in certain quantum mechanical scenario. Here we shall show how to associate a proof of Kochen-Specker theorem with a quantum pigeonhole effect and vise versa, e.g., from state-independent proofs of Kochen-Specker theorem some kind of state-independent quantum pigeonhole effects can be demonstrated. In particular, a state-independent version of the quantum Cheshire cat, which can be rendered as a kind of …
One fascinating way of revealing the quantum nonlocality is the all-versus-nothing test due to Greenberger, Horne, and Zeilinger (GHZ) known as GHZ paradox. So far genuine multipartite and multilevel GHZ paradoxes are known to exist only in systems containing an odd number of particles. Here we shall construct GHZ paradoxes for an arbitrary number (greater than 3) of particles with the help of qudit graph states on a special kind of graphs, called as GHZ graphs. Based on the GHZ paradox arising from a GHZ graph, we derive a Bell inequality with two $d$-outcome observables for each observer, wh…
Contextuality is a key signature of quantum non-classicality, which has been shown to play a central role in enabling quantum advantage for a wide range of information-processing and computational tasks. We study the logic of contextuality from a structural point of view, in the setting of partial Boolean algebras introduced by Kochen and Specker in their seminal work. These contrast with traditional quantum logic à la Birkhoff and von Neumann in that operations such as conjunction and disjunction are partial, only being defined in the domain where they are physically meaningful. We study how …
The discovery of quantum key distribution by Bennett and Brassard (BB84) bases on the fundamental quantum feature: incompatibility of measurements of quantum non-commuting observables. In 1991 Ekert showed that cryptographic key can be generated at a distance with help of entangled (correlated) quantum particles. Recently Barrett, Hardy and Kent showed that the non-locality used by Ekert is itself a good resource of cryptographic key even beyond quantum mechanics. Their result paved the way to new generation of quantum cryptographic protocols - secure even if the devices are built by the very …
How Quantum Contextuality disappears in the Classical Limit
Instituto de Matemática, Estatística e Computação Científica, Universidade Estadual de Campinas（坎皮纳斯州立大学数学、统计与科学计算研究所）；Perimeter Institute for Theoretical Physics（理论物理前沿研究所）
The discovery of quantum key distribution by Bennett and Brassard (BB84) bases on the fundamental quantum feature: incompatibility of measurements of quantum non-commuting observables. In 1991 Ekert showed that cryptographic key can be generated at a distance with help of entangled (correlated) quantum particles. Recently Barrett, Hardy and Kent showed that the non-locality used by Ekert is itself a good resource of cryptographic key even beyond quantum mechanics. Their result paved the way to new generation of quantum cryptographic protocols - secure even if the devices are built by the very …
The formalism of abstracted quantum mechanics is applied in a model of the generalized Liar Paradox. Here, the Liar Paradox, a consistently testable configuration of logical truth properties, is considered a dynamic conceptual entity in the cognitive sphere. Basically, the intrinsic contextuality of the truth-value of the Liar Paradox is appropriately covered by the abstracted quantum mechanical approach. The formal details of the model are explicited here for the generalized case. We prove the possibility of constructing a quantum model of the m-sentence generalizations of the Liar Paradox. T…
Quantum pre- and post-selection (PPS) paradoxes occur when counterfactual inferences are made about different measurements that might have been performed, between two measurements that are actually performed. The 3 box paradox is the paradigm example of such a paradox, where a ball is placed in one of three boxes and it is inferred that it would have been found, with certainty, both in box 1 and in box 2 had either box been opened on their own. Precisely what is at stake in PPS paradoxes has been unclear, and classical models have been suggested which are supposed to mimic the essential featur…
We present a logical type of proof of contextuality for a two-qubit state. We formulate a paradox that cannot be verified by a two-qubit system with local measurements while it is possible by using entanglement measurements. With our scheme we achieve $p_{\rm Hardy} \approx 0.167$, which is the highest probability obtained for a system of similar dimension. Our approach uses graph theory and the global exclusivity principle to give an interpretation of logical type of proofs of quantum correlations. We review the Hardy paradox and find connection to the KCBS inequality. We apply the same metho…
Most of the paradoxical, for the classical intuition, features of quantum theory were formulated for situations which involve a fixed number of particles. While one can now find a formulation of Bell's theorem for quantum fields, a Kochen-Specker-type reasoning is usually formulated for just one particle, or like in the case of Peres-Mermin square for two. A question emerges. Is it possible to formulate a contextuality proof for situation in which the numbers of particles are fundamentally undefined? We address this problem for bosonic fields. We introduce a representation of the $\mathfrak{su…
Empirical evidence has confirmed that quantum effects occur frequently also outside the microscopic domain, while quantum structures satisfactorily model various situations in several areas of science, including biological, cognitive and social processes. In this paper, we elaborate a quantum mechanical model which faithfully describes the 'Ellsberg paradox' in economics, showing that the mathematical formalism of quantum mechanics is capable to represent the 'ambiguity' present in this kind of situations, because of the presence of 'contextuality'. Then, we analyze the data collected in a con…
This thesis reports progress in the analysis of causality and multi-agent logical paradoxes in quantum and post-quantum theories. These research areas are highly relevant for the foundations of physics as well as the development of quantum technologies. In the first part, focussing on causality, we develop techniques for using generalised entropies to analyse distinctions between classical and non-classical causal structures. We derive new properties of Tsallis entropies of systems that follow from the relevant causal structure, and apply these to obtain new necessary constraints for classical…
Quantum non-locality tests have been of interest since the 1960's paper by Bell on the original EPR paradox.The present paper discusses whether the CGLMP (Bell) inequalities are possible tests for showing that quantum theory is not underpinned by local hidden variable theory (LHVT). It is found by applying Fine's theorem that the CGLMP approach involves a LHVT for the joint probabilities associated with the measurement of one observable from each of the two sub-systems, even though the underlying probabilities for joint measurements of all four observables may involve a non-local HVT. The latt…
We understand emergent quantum mechanics in the sense that quantum mechanics describes processes of physical emergence relating an assumed sub-quantum physics to macroscopic boundary conditions. The latter can be shown to entail top-down causation, in addition to usual bottom-up scenarios. With this example it is demonstrated that definitions of "realism" in the literature are simply too restrictive. A prevailing manner to define realism in quantum mechanics is in terms of pre-determination independent of the measurement. With our counter-example, which actually is ubiquitous in emergent, or s…
