Classical probabilistic models of (noisy) quantum systems are not only relevant for understanding the non-classical features of quantum mechanics, but they are also useful for determining the possible advantage of using quantum resources for information processing tasks. A common feature of these models is the presence of inaccessible information, as captured by the concept of preparation contextuality: There are ensembles of quantum states described by the same density operator, and hence operationally indistinguishable, and yet in any probabilistic (ontological) model, they should be described by distinct probability distributions. In this work, we quantify the inaccessible information of a model in terms of the maximum distinguishability of probability distributions associated to any pair of ensembles with identical density operators, as quantified by the total variation distance of the distributions. We obtain a family of lower bounds on this maximum distinguishability in terms of experimentally measurable quantities. In the case of an ideal qubit this leads to a lower bound of, approximately, 0.07. These bounds can also be interpreted as a new class of robust preparation non-contextuality inequalities. Our non-contextuality inequalities are phrased in terms of generalizations of max-relative entropy and trace distance for general operational theories, which could be of independent interest.
Under sufficiently strong noise any quantum system becomes preparation non-contextual, i.e., can be described by models with zero inaccessible information. Using our non-contextuality inequalities, we show that this can happen only if the noise channel has the average gate fidelity less than or equal to 1/D(1+1/2+...+1/D), where D is the dimension of the Hilbert space.
The Suppes-Zanotti inequalities involving the joint expectations of just three binary quantum observables are (re-)derived by the hull computation of the respective correlation polytope. A min-max calculation reveals its maximal quantum violations correspond to a generalized Tsirelson bound. Notions of "contextuality" motivated by such violations are critically reviewed.
We show that the perfect commutation graph is the sufficient tight condition for admitting the noncontextual description of each observable set satisfying it in the yes-no question scenario. With this condition, we propose a method for proving the monogamy relation between two informationtheoretic contextuality inequalities by decomposing the total commutation graph into perfect subgraphs. The results offer a powerful tool to investigate the contextuality and to understand quantum information theory. This theoretical work can be experimentally verified in current laboratorial technology.
Contextuality is one way of capturing the non-classicality of quantum theory. The contextual nature of a theory is often witnessed via the violation of non-contextuality inequalities---certain linear inequalities involving probabilities of measurement events. Using the exclusivity graph approach (one of the two main graph theoretic approaches for studying contextuality), it was shown [PRA 88, 032104 (2013); Annals of mathematics, 51-299 (2006)] that a necessary and sufficient condition for witnessing contextuality is the presence of an odd number of events (greater than three) which are either…
Two fundamental quantum resources, nonlocality and contextuality, can be connected through Bell inequalities that are violated by state-independent contextuality (SI-C) sets. These Bell inequalities allow for applications that require simultaneous nonlocality and contextuality. However, for existing Bell inequalities, the nonlocality produced by SI-C sets is very sensitive to noise. This precludes experimental implementation. Here we identify the Bell inequalities for which the nonlocality produced by SI-C sets is optimal, i.e., maximally robust to either noise or detection inefficiency, for t…
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…
Ilya Schmelzer wrote recently: {\it Nieuwenhuizen argued that there exists some "contextuality loophole" in Bell's theorem. This claim in unjustified}. It is made clear that this arose from attaching a meaning to the title and the content of the paper different from the one intended by Nieuwenhuizen. "Contextual loophole" means only that if the supplementary parameters describing measuring instruments are correctly introduced, Bell and Bell-type inequalities may not be proven. It is also stressed that a hidden variable model suffers from a "contextuality loophole" if it tries to describe diffe…
In this paper we first analyse Leggett and Garg's argument to the effect that macroscopic realism contradicts quantum mechanics. After making explicit all the assumptions in Leggett and Garg's reasoning, we argue against the plausibility of their auxiliary assumption of non-invasive measurability, using Bell's construction of stochastic pilot-wave theories as a counterexample. Violations of the Leggett-Garg inequality thus do not provide a good argument against macrorealism per se. We then apply Dzhafarov and Kujala's analysis of contextuality in the presence of signalling to the case of the L…
Nowadays contextuality is the hotest topic of quantum foundations and, especially, foundations of quantum information theory. This notion is characterized by the huge diversity of approaches and interpretations. One of the strongest trends in contextual research is to identify contextuality with violation of the Bell inequalities. We call this sort of contextuality {\it Bell test contextuality} (BTC). In this note, we criticize the BTC-approach. It can be compared with an attempt to identify the complex and theoretically nontrivial notion of randomness with a test for randomness (or a batch of…
We show that there are Bell-type inequalities for noncontextual theories that are violated by any quantum state. One of these inequalities between the correlations of compatible measurements is particularly suitable for testing this state-independent violation in an experiment.
We investigate the violation of non-contextuality by a class of continuous variable states, including variations of entangled coherent states (ECS's) and a two-mode continuous superposition of coherent states. We generalise the Kochen-Specker (KS) inequality discussed in A. Cabello, Phys. Rev. Lett. {\bf 101}, 210401 (2008) by using effective bidimensional observables implemented through physical operations acting on continuous variable states, in a way similar to an approach to the falsification of Bell-CHSH inequalities put forward recently. We test for state-independent violation of KS ineq…
Recent experiments have shown that nature violates noncontextual inequalities regardless of the state of the physical system. So far, all these inequalities involve measurements of dichotomic observables. We show that state-independent quantum contextuality can also be observed in the correlations between measurements of observables with genuinely continuous spectra, highlighting the universal character of the effect.
Quantum contextual sets have been recognized as resources for universal quantum computation, quantum steering and quantum communication. Therefore, we focus on engineering the sets that support those resources and on determining their structures and properties. Such engineering and subsequent implementation rely on discrimination between statistics of measurement data of quantum states and those of their classical counterparts. The discriminators considered are inequalities defined for hypergraphs whose structure and generation are determined by their basic properties. The generation is inhere…
We experimentally test quantum contextuality of a single qutrit using NMR. The contextuality inequalities based on nine observables developed by Kurzynski et. al. are first reformulated in terms of traceless observables which can be measured in an NMR experiment. These inequalities reveal the contextuality of almost all single-qutrit states. We demonstrate the violation of the inequality on four different initial states of a spin-1 deuterium nucleus oriented in a liquid crystal matrix, and follow the violation as the states evolve in time. We also describe and experimentally perform a single-s…
Quantum theory has the intriguing feature that is inconsistent with noncontextual hidden variable models, for which the outcome of a measurement does not depend on which other compatible measurements are being performed concurrently. While various proofs of such contextual behavior of quantum systems have been established, relatively little is known concerning the possibility to demonstrate this intriguing feature for indistinguishable particles. Here, we show in a simple and systematic manner that with projective measurements alone, it is possible to demonstrate quantum contextuality for such…
Contextuality means non-existence of a joint distribution for random variables recorded under mutually incompatible conditions, subject to certain constraints imposed on how the identity of these variables may change across these conditions. In simple quantum systems contextuality is indicated by violations of Bell-type or Leggett-Garg-type inequalities. These inequalities, however, are predicated on the assumption of no-signaling, defined as invariance of the distributions of measurement results with respect to other (e.g., earlier in time) measurements' settings. Signaling makes the inequali…
It has been argued that any test of quantum contextuality is nullified by the fact that perfect orthogonality and perfect compatibility cannot be achieved in finite precision experiments. We introduce experimentally testable two-qutrit violations of inequalities for noncontextual theories in which compatibility is guaranteed by the fact that measurements are performed on separated qutrits. The inequalities are inspired by the basic building block of the Kochen-Specker proof of quantum contextuality for a qutrit, despite the fact that their proof is completely independent of it.
Cyclic systems of dichotomous random variables have played a prominent role in contextuality research, describing such experimental paradigms as the Klyachko-Can-Binicoglu-Shumovky, Einstein-Podolsky-Rosen-Bell, and Leggett-Garg ones in physics, as well as conjoint binary choices in human decision making. Here, we understand contextuality within the framework of the Contextuality-by-Default (CbD) theory, based on the notion of probabilistic couplings satisfying certain constraints. CbD allows us to drop the commonly made assumption that systems of random variables are consistently connected. C…
Characterising unknown quantum states and measurements is a fundamental problem in quantum information processing. In this Letter, we provide a novel scheme to self-test local quantum systems using non-contextuality inequalities. Our work leverages the graph-theoretic framework for contextuality introduced by Cabello, Severini, and Winter, combined with tools from mathematical optimisation that guarantee the unicity of optimal solutions. As an application, we show that the celebrated Klyachko-Can-Biniciogglu-Shumovsky inequality and its generalisation to contextuality scenarios with odd n-cycl…
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…
This paper illustrates a direct connection between quantum steering and non-trivial preparation contextuality. In two party-two measurement per party-two outcomes per measurement $(2-2-2)$ Bell scenario, any argument of Bell nonlocality is a proof of trivial preparation contextuality; however, the converse may not hold. If one of the parties (say, Alice) performs the measurements of more than two dichotomic observables, then it is possible to find a set of non-trivial functional relations between Alice's observables. We argue that the existence of a suitable set of such non-trivial relations b…
Kochen-Specker (KS) theorem reveals the inconsistency between quantum theory and any putative underlying model of it satisfying the constraint of KS-noncontextuality. A logical proof of the KS theorem is one that relies only on the compatibility relations amongst a set of projectors (a KS set) to witness this inconsistency. These compatibility relations can be represented by a hypergraph, referred to as a contextuality scenario. Here we consider contextuality scenarios that we term KS-uncolourable, e.g., those which appear in logical proofs of the KS theorem. We introduce a hypergraph framewor…
The contextuality of quantum mechanics, i.e. the measurement outcome dependence upon previously made measurements, can be shown by the violation of inequalities based on measurements of well chosen observables. An important property of such observables is that their expectation value can be expressed in terms of probabilities of obtaining two exclusive outcomes. In order to satisfy this, inequalities have been constructed using either observables with a dichotomic spectrum or using periodic functions obtained from displacement operators in phase space. Here we identify the general conditions o…
The contextuality and noncontextuality notions are considered in framework of probability representation of quantum states. Example of qutrit states and violation of the noncontextuality inequalities are presented by using the spin tomogram and tomographic symbols of the observables.
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 describe a mathematical language for determining all possible patterns of contextuality in the dependence of stochastic outputs of a system on its deterministic inputs. The central notion is that of all possible couplings for stochastically unrelated outputs indexed by mutually incompatible values of inputs. A system is characterized by a pattern of which outputs can be "directly influenced" by which inputs (a primitive relation, hypothetical or normative), and by certain constraints imposed on the outputs (such as Bell-type inequalities or their quantum analogues). The set of couplings com…
One of the interesting topics in quantum contextuality is the construction for various non-contextual inequalities. By introducing a new structure called hyper-graph, we present a general method, which seems to be analytic and extensible, to derive the non-contextual inequalities for the qutrit systems. Based on this, several typical families of non-contextual inequalities are discussed. And our approach may also help us to simplify some state-independent proofs for quantum contextuality in one of our recent works.
The violation of Bell, CHSH and CH inequalities indicates only that the assumption of "conterfactual definiteness" and/or the probabilistic models used in proofs were incorrect. In this paper we discuss in detail an intimate relation between experimental protocols and probabilistic models. In particular we show that local realistic and stochastic hidden variable models are inconsistent with the experimental protocols used in spin polarization correlation experiments. In particular these models neglect a contextual character of quantum theory (QT) and do not describe properly quantum measuremen…
Violations of Bell inequalities have been an incontestable indicator of non-classicality since the seminal paper by John Bell. However, recent claims of Bell inequalities violations with classical light have cast some doubts on their significance as hallmarks of non-classicality. Here, we challenge those claims. The crux of the problem is that such classical experiments simulate quantum probabilities with intensities of classical fields. However, fields intensities measurements are radically different from single-photon detections, which are primitives of any genuine Bell experiment. We show t…
Quantum contextuality is a source of quantum computational power and a theoretical delimiter between classical and quantum structures. It has been substantiated by numerous experiments and prompted generation of state independent contextual sets, that is, sets of quantum observables capable of revealing quantum contextuality for any quantum state of a given dimension. There are two major classes of state-independent contextual sets - the Kochen-Specker ones and the operator-based ones. In this paper, we present a third, hypergraph-based class of contextual sets. Hypergraph inequalities serve a…
Violation of a noncontextuality inequality or the phenomenon referred to `quantum contextuality' is a fundamental feature of quantum theory. In this article, we derive a novel family of noncontextuality inequalities along with their sum-of-squares decompositions in the simplest (odd-cycle) sequential-measurement scenario capable to demonstrate Kochen-Specker contextuality. The sum-of-squares decompositions allow us to obtain the maximal quantum violation of these inequalities and a set of algebraic relations necessarily satisfied by any state and measurements achieving it. With their help, we …
The development of Noisy Intermediate Scale Quantum Computers (NISQC) provides for the Quantum Information community new tools to perform quantum experiences from an individual laptop. It facilitates interdisciplinary research in the sense that theoretical descriptions of properties of quantum physics can be translated to experiments easily implementable on a NISCQ. In this note I test large state-independent inequalities for quantum contextuality on finite geometric structures encoding the commutation relations of the generalized N-qubit Pauli group. The bounds predicted by Non-Contextual Hid…
The study of ontology (hidden variables) provides for a vital ground on which significant non-classical features of quantum theory are revealed. One such non-classical ontic-feature is preparation contextuality (PC) and advantage in oblivious communication tasks is one of its operational signatures. This article primarily pursues the ontic-feature underlying quantum advantage in communication complexity (CC). We construct oblivious communication tasks tailored to given CC problems. We upper-bound the classical success probability of these oblivious communication tasks, obtaining preparation no…
Nonlocality inequalities for multi-party systems act as contextuality inequalities for single qudit systems of suitable dimensions. In this paper, we propose the procedure for adaptation of nonlocality-based quantum conferencing protocols (QCPs) to contextuality-based QCPs. Unlike the nonlocality-based protocols, the proposed protocols do not involve nonlocal states. These protocols can also be implemented in a class of noisy channels without any compromise in the key generation rate. As an illustration of the procedure, we present a QCP based on Mermin's contextuality inequality. As a signifi…
The information-theoretic approach to Bell's theorem is developed with use of the conditional $q$-entropies. The $q$-entropic measures fulfill many similar properties to the standard Shannon entropy. In general, both the locality and noncontextuality notions are usually treated with use of the so-called marginal scenarios. These hypotheses lead to the existence of a joint probability distribution, which marginalizes to all particular ones. Assuming the existence of such a joint probability distribution, we derive the family of inequalities of Bell's type in terms of conditional $q$-entropies f…
The finite set of subsystems of a finite quantum system with variables in ${\mathbb Z}(n)$, is studied as a Heyting algebra. The physical meaning of the logical connectives is discussed. It is shown that disjunction of subsystems is more general concept than superposition. Consequently the quantum probabilities related to projectors in the subsystems, are incompatible with associativity of the join in the Heyting algebra, unless if the variables belong to the same chain. This leads to contextuality, which in the present formalism has as contexts, the chains in the Heyting algebra. Logical Bell…
We unify the resource-theoretic and the cohomological perspective on quantum contextuality. At the center of this unification stands the notion of the contextual fraction. For both symmetry and parity based contextuality proofs, we establish cohomological invariants which are witnesses of state-dependent contextuality. We provide two results invoking the contextual fraction, namely (i) refinements of logical contextuality inequalities, and (ii) upper bounds on the classical cost of Boolean function evaluation, given the contextual fraction of the corresponding measurement-based quantum computa…
Cabello-Severini-Winter and Abramsky-Hardy (building on the framework of Abramsky-Brandenburger) both provide classes of Bell and contextuality inequalities for very general experimental scenarios using vastly different mathematical techniques. We review both approaches, carefully detail the links between them, and give simple, graph-theoretic methods for finding inequality-free proofs of nonlocality and contextuality and for finding states exhibiting strong nonlocality and/or contextuality. Finally, we apply these methods to concrete examples in stabilizer quantum mechanics relevant to unders…
Contextuality provides one of the fundamental characterizations of quantum phenomena, and can be used as a resource in lots of quantum information processing. In this paper, we summarize and derive some equivalent noncontextual inequalities from different noncontextual models of the proofs for Kochen-Specker theorem based on Greenberger-Horne-Zeilinger states. These noncontextual inequalities are equivalent up to some correlation items which hold both for noncontextual hidden variable theories and quantum mechanics. Therefore, using single-photon hyperentangled Greenberger-Horne-Zeilinger stat…
Recent experiments have demonstrated ququart state-independent quantum contextuality and qutrit state-dependent quantum contextuality. So far, the most basic form of quantum contextuality pointed out by Kochen and Specker, and Bell, has eluded experimental confirmation. Here we present an experimentally feasible test to observe qutrit state-independent quantum contextuality using single photons in a three-path setup. In addition, we show that if the same measurements are performed on two entangled qutrits, rather than sequentially on the same qutrit, then the noncontextual inequality becomes a…
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.
Contextuality and entanglement are valuable resources for quantum computing and quantum information. Bell inequalities are used to certify entanglement; thus, it is important to understand why and how they are violated. Quantum mechanics and behavioral sciences teach us that random variables measuring the same content (the answer to the same Yes or No question) may vary, if measured jointly with other random variables. Alice and Bob raw data confirm Einsteinian non-signaling, but setting dependent experimental protocols are used to create samples of coupled pairs of distant outcomes and to est…
We show that the noncontextual inequality proposed by Klyachko et al. [Phys. Rev. Lett. 101, 020403 (2008)] belongs to a broader family of inequalities, one associated to each compatibility structure of a set of events (a graph), and its independence number. These have the surprising property that the maximum quantum violation is given by the Lovasz theta-function of the graph, which was originally proposed as an upper bound on its Shannon capacity. Furthermore, probabilistic theories beyond quantum mechanics may have an even larger violation, which is given by the so-called fractional packing…
Quantum nonlocality can be revealed "via local contextuality" in qudit-qudit entangled systems with $d > 2$, that is, through the violation of inequalities containing Alice-Bob correlations that admit a local description, and Alice-Alice correlations (between the results of sequences of measurements on Alice's subsystem) that admit a local (but contextual) description. A fundamental question to understand the respective roles of entanglement and local contextuality is whether nonlocality via local contextuality exists when the parties have only qubit-qubit entanglement. Here we respond affirma…
We explore non-classical correlations in n-cycle setting. In particular, we focus on correlations manifested by Kochen-Specker-Klyachko box (KS box), scenarios involving n-cycle non-contextuality inequalities and Popescu-Rohlrich boxes (PR box). We provide the criteria for optimal classical simulation of a KS box of arbitrary n dimension. The non-contextuality inequalities are analysed for n-cycle setting, and the condition for the quantum violation for odd as well as even n-cycle is discussed. We offer a simple extension of even cycle non-contextuality inequalities to the continuous variable …
This work develops analytic methods to quantitatively demarcate quantum reality from its subset of classical phenomenon, as well as from the superset of general probabilistic theories. Regarding quantum nonlocality, we discuss how to determine the quantum limit of Bell-type linear inequalities. In contrast to semidefinite programming approaches, our method allows for the consideration of inequalities with abstract weights, by means of leveraging the Hermiticity of quantum states. Recognizing that classical correlations correspond to measurements made on separable states, we also introduce a pr…
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 …
The derivation of Bell inequalities for beables is well-known to require a "no-conspiracy" assumption. This assumption is widely accepted, the alternative being correlations between instrument settings and hidden beables. Two further assumptions are identified here: (1) a "no-contextuality" assumption, similar to the prohibition of contextuality that is required to derive the Kochen-Specker theorem, which is closely related to the "no-conspiracy" assumption; (2) a "no-correlation" assumption, which prohibits correlations between hidden beables. The three assumptions together are less acceptabl…
In this work we analyse the notion of measurement non-contextuality (MNC) and identify contextual scenarios which involve sequential measurements of only a single measurement device. We show that any non-contextual ontological model fails to explain the statistics of outcomes of a single carefully constructed positive operator valued measure (POVM) executed sequentially on a quantum system. The context of measurement arises from the different configurations in which the device can be used. We develop an inequality from the non-contextual (NC) ontic model, and construct a quantum situation invo…
