We discuss the problem of hidden variables and the motivation for introducting them in quantum mechanics. These include determinism, and the problem of meassurement and incompleteness. We first discuss Von-Neumann's imposisbility proof and then analyse it's weakness in terms of Bell's explicit hidden variable model of spin one-half particles. We next discuss Gleason's theorem and Kochen-Specker theorem and bring out the troublems with non contextual hidden variable theories. An important role is played by Einstein locality in the discussion of hidden variable theories as was first brought out by Einstein, Podolsky and Rosen. We elaborate it's various implications such as Bell's theorem in terms of Bell's inequalities as well as later work in which Bell's theorem follows without using inequalities.
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…
Contextuality is often referred to as a generalization of non-locality. In this work, using the hypergraph approach for contextuality we show how to associate a contextual scenario to a general k-partite non local game, and consider the reverse direction: how and when is it possible to represent a general contextuality scenario as a non local game. Using the notion of conditional contextuality, we show that it is possible to embed any contextual scenario in a two players non local game. We also discuss different equivalences of contextuality scenarios and show that the construction used in the…
The author cannot grant that Quantum Mechanics satisfies the axioms described here when repeated measurements are at stake. This does not compromise the validity of the conclusions of the previous article "A glance beyond the quantum model", Proc. Roy. Soc. Lond. A 466, 881-890 (2009).
We discuss quantum non-locality and contextuality using the notion of transition sets. This approach provides a way to obtain a direct logical contradiction with locality/non-contextuality in the EPRB gedanken experiment as well as a clear graphical illustration of what violations of Bell inequalities quantify. In particular, we show graphically how these violations are related to measures of non-local transition sets. We also introduce a new form of contextuality, {\em measurement ordering contextuality}, i.e. there exists commuting operators $\hat{\mathcal{A}}$ and $\hat{\mathcal{B}}$ such t…
In [Phys. Rev. Lett. 114, 250401 (2015)] the sharing of non-locality by multiple observers was demonstrated through the quantum violation of Clauser-Horne-Shimony-Halt inequality. In this paper we provide a scheme for sharing of non-locality and non-trivial preparation contextuality sequentially through the quantum violation of a family of Bell's inequalities where Alice and Bob perform $2^{n-1}$ and $n$ numbers of measurements of dichotomic observables respectively. For this, we consider that Alice always performs projective measurement and multiple Bobs sequentially perform unsharp measureme…
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…
We highlight the existence of a joint probability distribution as the common underpinning assumption behind Bell-type, contextuality, and Leggett-Garg-type tests. We then present a procedure to translate contextual scenarios into temporal Leggett-Garg-type and spatial Bell-type ones. To demonstrate the generality of this approach we construct a family of spatial Bell-type inequalities. We show that in Leggett-Garg scenario a necessary condition for contextuality in time is given by a violation of consistency conditions in Consistent Histories approach to quantum mechanics.
Bell non-locality and Kochen-Specker (KS) contextuality are logically independent concepts, fuel different protocols with quantum vs classical advantage, and have distinct classical simulation costs. A natural question is what are the relations between these concepts, advantages, and costs. To address this question, it is useful to have a map that captures all the connections between Bell non-locality and KS contextuality in quantum theory. The aim of this work is to introduce such a map. After defining the theory-independent notions of Bell non-locality and KS contextuality for ideal measurem…
A unified view on the phenomenon of monogamy exhibited by Bell inequalities and non-contextuality inequalities arising from the no-signaling and no-disturbance principles is presented using the graph-theoretic method introduced in \textit{Phys. Rev. Lett. 109, 050404 (2012)}. We propose a novel type of trade-off, namely Bell inequalities that do not exhibit monogamy features of their own can be activated to be monogamous by the addition of a local contextuality term. This is illustrated by means of the well-known $\mathcal{I}_{3322}$ inequality, and reveals a resource trade-off between biparti…
We examine the locality assumption of Bell's theorem in three steps of EPRB experiment. Depending on the context, locality is embodied in the conditions of separability, local causality, factorizability, relativistic causality, and non-contextuality. We show that separability, characterized by the constraint of zero covariance, is equivalent to local causality which leads to factorizability through the measurement process. Factorizability is the conjunction of parameter independence and outcome independence. It is commonly believed that relativistic causality is equivalent to parameter indepen…
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 st…
A PhD student is locked inside a box, imitating a quantum system by mimicking the measurement statistics of any viable observable nominated by external observers. Inside a second box lies a genuine quantum system. Either box can be used to pass a test for contextuality - and from the perspective of an external observer, be operationally indistinguishable. There is no way to discriminate between the two boxes based on the output statistics of any contextuality test. This poses a serious problem for contextuality tests to be used as viable tests for device independent quantumness, and severely l…
The question of a hidden variable interpretation of quantum contextuality in the Mermin-Peres square is considered. The Kochen-Specker theorem implies that quantum mechanics may be interpreted as a contextual hidden variable theory. It is shown that such a hidden variable description can be viewed as either contextual in the random variables mapping hidden states to observable outcomes or in the probability measure on the hidden state space. The latter view suggests that this apparent contextuality may be interpreted as a simple consequence of measurement disturbance, wherein the initial hidde…
An experiment has recently been performed to demonstrate quantum nonlocality by establishing contextuality in one of a pair of photons encoding four qubits; however, low detection efficiencies and use of the fair-sampling hypothesis leave these results open to possible criticism due to the detection loophole. In this Letter, a physically motivated local hidden-variable model is considered as a possible mechanism for explaining the experimentally observed results. The model, though not intrinsically contextual, acquires this quality upon post-selection of coincident detections.
Quantum mechanics allows only certain sets of experimental results (or "probabilistic models") for Bell-type quantum non-locality experiments. A derivation of this set from simple physical or information theoretic principles would represent an important step forward in our understanding of quantum mechanics, and this problem has been intensely investigated in recent years. "Macroscopic locality," which requires the recovery of locality in the limit of large numbers of trials, is one of several principles discussed in the literature that place a bound on the set of quantum probabilistic models.…
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 …
We use the mathematical language of sheaf theory to give a unified treatment of non-locality and contextuality, in a setting which generalizes the familiar probability tables used in non-locality theory to arbitrary measurement covers; this includes Kochen-Specker configurations and more. We show that contextuality, and non-locality as a special case, correspond exactly to obstructions to the existence of global sections. We describe a linear algebraic approach to computing these obstructions, which allows a systematic treatment of arguments for non-locality and contextuality. We distinguish a…
Random numbers represent an indispensable resource for many applications. A recent remarkable result is the realization that non-locality in quantum mechanics can be used to certify genuine randomness through Bell's theorem, producing reliable random numbers in a device independent way. Here, we explore the contextuality aspect of quantum mechanics and show that true random numbers can be generated using only single qutrit (three-state systems) without entanglement and non-locality. In particular, we show that any observed violation of the Klyachko-Can-Binicioglu-Shumovsky (KCBS) inequality [P…
This paper deals with three traditional ways of defining contextuality: (C1) in terms of (non)existence of certain joint distributions involving measurements made in several mutually exclusive contexts; (C2) in terms of relationship between factual measurements in a given context and counterfactual measurements that could be made if one used other contexts; and (C3) in terms of (non)existence of ``hidden variables'' that determine the outcomes of all factually performed measurements. It is generally believed that the three meanings are equivalent, but the issues involved are not entirely trans…
Two distant systems can exhibit quantum nonlocality even though the correlations between them admit a local model. This nonlocality can be revealed by testing extra correlations between successive measurements on one of the systems which do not admit a noncontextual model whatever the reduced state of this system is. This shows that quantum contextuality plays a fundamental role in quantum nonlocality, and allows an experimental test of the Kochen-Specker with locality theorem.
A primary goal in recent research on contextuality has been to extend this concept to cases of inconsistent connectedness, where observables have different distributions in different contexts. This article proposes a solution within the framework of probabilistic causal models, which extend hidden-variables theories, and then demonstrates an equivalence to the contextuality-by-default (CbD) framework. CbD distinguishes contextuality from direct influences of context on observables, defining the latter purely in terms of probability distributions. Here we take a causal view of direct influences…
Wheeler's {\it observer-participancy} and the related {\it it from bit} credo refer to quantum non-locality and contextuality. The mystery of these concepts slightly starts unveiling if one encodes the (in)compatibilities between qubit observables in the relevant finite geometries. The main objective of this treatise is to outline another conceptual step forward by employing Grothendieck's {\it dessins d'enfants} to reveal the topological and (non)algebraic machinery underlying the measurement acts and their information content.
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 c…
Considering an extended type of Bohm's version of EPR thought experiment, we derive Bell's inequality for the case of factorizable contextual hidden variable theories which are consistent with the predictions of quantum theory. Usually factorizability is associated with non-contextuality. Here, we show that factorizability is consistent with contextuality, even for the ordinary Bohm's version of EPR thought experiment.
Contextuality has been conjectured to be a super-classical resource for quantum computation, analogous to the role of non-locality as a super-classical resource for communication. We show that the presence of contextuality places a lower bound on the amount of classical memory required to simulate any quantum sub-theory, thereby establishing a quantitative connection between contextuality and classical simulability. We apply our result to the qubit stabilizer sub-theory, where the presence of state-independent contextuality has been an obstacle in establishing contextuality as a quantum comput…
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.
Bosonic bunching occurs within quantum physics and can be mimicked classically by noncontextual hidden-variable models, which excludes this phenomenon as a means to prove stronger-than-quantum contextuality.
The Kochen--Specker (KS) theorem reveals the nonclassicality of single quantum systems. In contrast, Bell's theorem and entanglement concern the nonclassicality of composite quantum systems. Accordingly, unlike incompatibility, entanglement and Bell non-locality are not necessary to demonstrate KS-contextuality. However, here we find that for multiqubit systems, entanglement and non-locality are both essential to proofs of the Kochen--Specker theorem. Firstly, we show that unentangled measurements (a strict superset of local measurements) can never yield a logical (state-independent) proof of …
The Kochen-Specker theorem states that noncontextual hidden variable models are inconsistent with the quantum predictions for every yes-no question on a qutrit, corresponding to every projector in three dimensions. It has been suggested [D. A. Meyer, Phys. Rev. Lett. 83, 3751 (1999)] that the inconsistency would disappear when we are restricted to projectors on unit vectors with rational components; that noncontextual hidden variables could reproduce the quantum predictions for rational vectors. Here we show that a qutrit state with rational components violates an inequality valid for nonconte…
We argue that for a \emph{single particle} Bell's inequality is a consequence of noncontextuality and is \emph{incompatible} with statistical predictions of quantum mechanics. Thus noncontextual models can be empirically falsified, \emph{independent} of locality condition. For this an appropriate entanglement between \emph{disjoint} Hilbert spaces pertaining to translational and spin degrees of freedom of a single spin-1/2 particle is invoked
Contextuality is a natural generalization of nonlocality which does not need composite systems or spacelike separation and offers a wider spectrum of interesting phenomena. Most notably, in quantum mechanics there exist scenarios where the contextual behavior is independent of the quantum state. We show that the quest for an optimal inequality separating quantum from classical noncontextual correlations in an state-independent manner admits an exact solution, as it can be formulated as a linear program. We introduce the noncontextuality polytope as a generalization of the locality polytope, an…
Quantum mechanics implies that not all physical properties can be simultaneously well defined, such as the momentum and position due to Heisenberg uncertainty principle. Some alternative theories have been explored, notably the non-contextual hidden variable theories in which the properties of a system have pre-defined values which are independent of the measurement contextual. However, the Kochen-Specker theorem showed that such non-contextual hidden variable theories are in conflict with quantum mechanics. Recently, a state-independent inequality satisfied by non-contextual hidden variable t…
Everyday experience supports the existence of physical properties independent of observation in strong contrast to the predictions of quantum theory. In particular, existence of physical properties that are independent of the measurement context is prohibited for certain quantum systems. This property is known as contextuality. This paper studies whether the process of decay in space-time generally destroys the ability of revealing contextuality. We find that in the most general situation the decay property does not diminish this ability. However, applying certain constraints due to the space-…
The Pauli groups are ubiquitous in quantum information theory because of their usefulness in describing quantum states and operations and their readily understood symmetry properties. In addition, the most well-understood quantum error correcting codes -- stabilizer codes -- are built using Pauli operators. The eigenstates of these operators -- stabilizer states -- display a structure (e.g., mutual orthogonality relationships) that has made them useful in examples of multi-qubit non-locality and contextuality. Here, we apply the graph-theoretical contextuality formalism of Cabello, Severini an…
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…
We report a single-neutron optical experiment to demonstrate the violation of a Bell-like inequality. Entanglement is achieved not between particles, but between the degrees of freedom, in this case, for a single-particle. The spin-{\small 1/2} property of neutrons are utilized. The total wave function of the neutron is described in a tensor product Hilbert space. A Bell-like inequality is derived not by a non-locality but by a contextuality. Joint measurements of the spinor and the path properties lead to the violation of a Bell-like inequality. Manipulation of the wavefunction in one Hilbert…
The quantum predictions for a single nonrelativistic spin-1/2 particle can be reproduced by noncontextual hidden variables. Here we show that quantum contextuality for a relativistic electron moving in a Coulomb potential naturally emerges if relativistic effects are taken into account. The contextuality can be identified through the violation of noncontextuality inequalities. We also discuss quantum contextuality for the free Dirac electron as well as the relativistic Dirac oscillator.
We present a new and feasible test proving quantum contextuality in four-dimensional Hiltbert space. In our scheme, a contradiction between quantum mechanics and noncontextual hidden variables is revealed through the measurement statistics of two joint measurements of an ensemble of non-entangled qubits. The origin of nonclassicality is fully attributed to the measurement process. Quantum optical realization using photon pairs and linear optics are presented.
In a noncontextual hidden variable model of quantum theory, hidden variables determine the outcomes of every measurement in a manner that is independent of how the measurement is implemented. Using a generalization of this notion to arbitrary operational theories and to preparation procedures, we demonstrate that a particular two-party information-processing task, "parity-oblivious multiplexing," is powered by contextuality in the sense that there is a limit to how well any theory described by a noncontextual hidden variable model can perform. This bound constitutes a "noncontextuality inequal…
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…
A theory is universal contextual if its prediction cannot be reproduced by an ontological model satisfying both preparation and measurement noncontextuality assumptions. In this report, we first generalize the logical proofs of quantum preparation and measurement contextuality for qubit system for any odd number of preparations and measurements. Based on the logical proof, we derive testable universally non-contextual inequalities violated by quantum theory. We then propose a class of two-party communication games and show that the average success probability of winning such games is solely li…
Since the enlightening proofs of quantum contextuality first established by Kochen and Specker, and also by Bell, various simplified proofs have been constructed to exclude the non-contextual hidden variable theory of our nature at the microscopic scale. The conflict between the non-contextual hidden variable theory and quantum mechanics is commonly revealed by Kochen-Specker (KS) sets of yes-no tests, represented by projectors (or rays), via either logical contradictions or noncontextuality inequalities in a state-(in)dependent manner. Here we first propose a systematic and programmable const…
One of the central foundational questions of physics is to identify what makes a system quantum as opposed to classical. One seminal notion of classicality of a quantum system is the existence of a non-contextual hidden variable model as introduced in the early work by Bell, Kochen and Specker. In quantum optics, the non-negativity of the Wigner function is a ubiquitous notion of classicality. In this work we establish an equivalence between these two concepts. In particular, we show that any non-contextual hidden variable model for Gaussian quantum optics has an alternative non-negative Wigne…
The question of whether quantum phenomena can be explained by classical models with hidden variables is the subject of a long lasting debate. In 1964, Bell showed that certain types of classical models cannot explain the quantum mechanical predictions for specific states of distant particles. Along this line, some types of hidden variable models have been experimentally ruled out. An intuitive feature for classical models is non-contextuality: the property that any measurement has a value which is independent of other compatible measurements being carried out at the same time. However, the res…
The Kochen-Specker theorem, Bell inequalities, and several other tests that were designed to rule out hidden-variable theories, assume the existence of observables having infinitely sharp eigenvalues. A paradigmatic example is spin-1/2. It is measured with a Stern-Gerlach array whose outputs are divided into two classes, spin-up and spin-down, in correspondence to the two spots observed on a detection screen. The spot's finite size is attributed to imperfections of the measuring device. This assumption turns the experimental output into a dichotomic, discrete one, thereby allowing the assignme…
Bell's seminal paper shows that some correlations in quantum theory are not reconcilable with hidden variables and the classical notion of locality. Yet, a weaker notion of locality, known as no-signalling, survives the no-go-result. Here, we study its restrictiveness by considering the full set of local quantum observables. This leads to a much larger set of no-signalling constraints than usually considered, which is enough to exclude PR-boxes and other types of super-quantum correlations. We emphasise the close connection with contextuality, in particular, we show how no-signalling arises na…
For spin-1/2 particles, using a suitable Mach-Zehnder-type setup with a spin-flipper, we argue that it is a direct consequence of the quantum mechanical treatment that an experimentally verifiable \textit{subensemble} mean of the measured values of an arbitrarily chosen spin variable exhibits dependence on the choice of a comeasurable `path' observable. This, in turn, enables inferring path-spin contextuality at the level of individual measured values of spin that are predetermined using a relevant hidden-variable model applied to our setup.
Indirect Indication of the KCBS-Type Quantum Contextuality by the CHSH-Bell Test
Faculty of Engineering and Natural Sciences, Sabanci University（萨贝里大学工程与自然科学学院）；Sabanci University（萨贝里大学）；International Centre for Theory of Quantum Technologies, University of Gdansk（格但斯克大学量子技术理论国际中心）；University of Gdansk（格但斯克大学）
