Noncontextuality Inequalities Research Articles

Lifting noncontextuality inequalities

Lifting noncontextuality inequalities

Single-Shot Readout of a Solid-State Electron Spin Qutrit.

Quantum systems usually feature a rich multilevel structure with promising resources for developing superior quantum technologies compared with their binary counterpart. Single-shot readout of these high-dimensional quantum systems is essential for exploiting their potential. Although there have been various high-spin systems, the single-shot readout of the overall state of high-spin systems remains a challenging issue. Here we demonstrate a reliable single-shot readout of spin qutrit state in a low-temperature solid-state system utilizing a binary readout scheme. We achieve a single-shot readout of an electron spin qutrit associated with a single nitrogen-vacancy center in diamond with an average fidelity of 87.80%. We use this spin qutrit system to verify quantum contextuality, a fundamental test of quantum mechanics. We observe a violation of the noncontextual hidden variable inequality with the developed single-shot readout in contrast to the conventional binary readout. These results pave the way for developing quantum information processing based on spin qutrits.

Coherence and contextuality in a Mach-Zehnder interferometer

We analyse nonclassical resources in interference phenomena using generalized noncontextuality inequalities and basis-independent coherence witnesses. We use recently proposed inequalities that witness both resources within the same framework. We also propose, in view of previous contextual advantage results, a systematic way of applying these tools to characterize advantage provided by coherence and contextuality in quantum information protocols. We instantiate this methodology for the task of quantum interrogation, famously introduced by the paradigmatic bomb-testing interferometric experiment, showing contextual quantum advantage for such a task.

Corrected Bell and non-contextuality inequalities for realistic experiments.

Contextuality is a feature of quantum correlations. It is crucial from a foundational perspective as a non-classical phenomenon, and from an applied perspective as a resource for quantum advantage. It is commonly defined in terms of hidden variables, for which it forces a contradiction with the assumptions of parameter-independence and determinism. The former can be justified by the empirical property of non-signalling or non-disturbance, and the latter by the empirical property of measurement sharpness. However, in realistic experiments neither empirical property holds exactly, which leads to possible objections to contextuality as a form of non-classicality, and potential vulnerabilities for supposed quantum advantages. We introduce measures to quantify both properties, and introduce quantified relaxations of the corresponding assumptions. We prove the continuity of a known measure of contextuality, the contextual fraction, which ensures its robustness to noise. We then bound the extent to which these relaxations can account for contextuality, via corrections terms to the contextual fraction (or to any non-contextuality inequality), culminating in a notion of genuine contextuality, which is robust to experimental imperfections. We then show that our result is general enough to apply or relate to a variety of established results and experimental set-ups. This article is part of the theme issue 'Quantum contextuality, causality and freedom of choice'.

Using a resource theoretic perspective to witness and engineer quantum generalized contextuality for prepare-and-measure scenarios

We employ the resource theory of generalized contextuality as a tool for analyzing the structure of prepare-and-measure scenarios. We argue that this framework simplifies proofs of quantum contextuality in complex scenarios and strengthens existing arguments regarding robustness of experimental implementations. As a case study, we demonstrate quantum contextuality associated with any nontrivial noncontextuality inequality for a class of useful scenarios by noticing a connection between the resource theory and measurement simulability. Additionally, we expose a formal composition rule that allows engineering complex scenarios from simpler ones. This approach provides insights into the noncontextual polytope structure for complex scenarios and facilitates the identification of possible quantum violations of noncontextuality inequalities.

Qubit measurements for testing contextuality through the violation of Liang-Spekkens-Wiseman-Yu-Oh inequality

Contextuality is a striking feature of our nature predicted by quantum theory and tested by various experiments. In a qubit system, testing contextuality requires scenarios distinct from traditional ones for systems with dimension of three or greater. In this paper, we consider the noncontextuality inequality developed by Liang-Spekkens-Wiseman and Yu-Oh, and investigate the set of three qubit measurements that can be used to test and violate this inequality. Our numerical results show that two of the three measurements can be almost arbitrary. Moreover, we have singled out several extremely interesting sets of measurements, e.g. one measurements can be white noise while the other two are arbitrarily close to white noise. These sets fulfill our knowledge about contextuality in this minimum quantum system.

Experimental Test of High-Dimensional Quantum Contextuality Based on Contextuality Concentration.

Contextuality is a distinctive feature of quantum theory and a fundamental resource for quantum computation. However, existing examples of contextuality in high-dimensional systems lack the necessary robustness required in experiments. Here, we address this problem by identifying a family of noncontextuality inequalities whose maximum quantum violation grows with the dimension of the system. At first glance, this contextuality is the single-system version of multipartite Bell nonlocality taken to an extreme form. What is interesting is that the single-system version achieves the same degree of contextuality but uses a Hilbert space of lower dimension. That is, contextuality "concentrates" as the degree of contextuality per dimension increases. We show the practicality of this result by presenting an experimental test of contextuality in a seven-dimensional system. By simulating sequences of quantum ideal measurements with destructive measurements and repreparation in an all-optical setup, we report a violation of 68.7 standard deviations of the simplest case of the noncontextuality inequalities identified. Our results advance the investigation of high-dimensional contextuality, its connection to the Clifford algebra, and its role in quantum computation.

Monogamy relations of entropic non-contextual inequalities and their experimental demonstration

We analyze monogamous relationships of abritrary entropic non-contextuality (ENC) inequalities via a graph theoretic approach. While ENC inequalities are important in quantum information theory, a theoretical and experimental understanding of their monogamous nature is still elusive. We analyze conditions for ENC inequalities to exhibit a monogamous relationship and derive the same explicitly for an entropic version of the Bell-CHSH scenario. Next, we consider two different sets of three qubit states and experimentally show that they exhibit monogamy of entropic Bell-CHSH inequality. The monogamous relationship of these inequalities is theoretically exhibited via an insightful proof and is experimentally demonstrated on an NMR quantum information processor. We also develop a novel and easy-to-implement experimental method to evaluate entropies on an NMR quantum information processor using only the information about expectation values of the observables.

Revealing quantum contextuality using a single measurement device

In this paper we analyze the notion of measurement noncontextuality (MNC) and identify contextual scenarios which involve sequential measurements of only a single measurement device. We show that any noncontextual 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 noncontextual (NC) ontic model, and construct a quantum situation involving measurements from the Klyachko-Can-Binicio\ifmmode \check{g}\else \v{g}\fi{}lu-Shumovsky (KCBS) inequality. We show that the resultant statistics arising from this device violate our NC inequality. This device can be generalized by incorporating measurements from arbitrary $n$-cycle contextuality inequalities of which $n=5$ corresponds to the KCBS inequality. We show that the NC and quantum bounds for various scenarios can be derived more easily using only the functional relationships between the outcomes for larger values $n$. This makes it one of the simpler contextual inequalities to analyze.

Quantum sets of the multicolored-graph approach to contextuality

The CHSH inequalities are the most famous examples of Bell inequalities. Cabello, Severini, and Winter (CSW) came up with a graph approach to noncontextuality inequalities, which connects some graph-theoretic concepts to quantum and classical correlations. For example, the theta body of the exclusivity graph can be associated with the set of correlations achieved by quantum theory. Following the CSW approach, one may think that the theta body of the CHSH graph, is equal to the quantum set of the CHSH Bell inequality, but is this really true? All assumptions about the CHSH inequalities come from Bell scenarios, while CSW approach only demands the exclusivity structure of a non-contextuality (NC) scenario. To deal with the extra structure related to the presence of different players in a Bell scenario like CHSH, the colored-graph approach was introduced. Does it make any difference to think about CHSH as a Bell scenario or a more general NC scenario? The Bell CHSH inequality is represented by a bicolored graph and the NC CHSH inequality by a simple graph, which is the shadow of the colored graph. In general, we have that the theta body of the colored graph is a subset of the theta body of its shadow graph in the same way that the Lov\`asz number, which corresponds to the quantum bound, of the simple graph is greater than or equal to the Lov\`asz number of the colored graph. In the case of the CHSH inequality, we have that the Lov\`asz numbers are equal. Does this accident also hold for the corresponding quantum sets? Is it true that the theta bodies are equal, which would mean that every correlation reached by quantum theory applied to the CHSH NC scenario could also be obtained at the in principle more restrictive CHSH Bell scenario? In this paper our answer to such a question is negativ. This implies that there are quantum correlations which can not be obtained under Bell restrictions.

Irreducible Magic Sets for n-Qubit Systems.

Magic sets of observables are minimal structures that capture quantum state-independent advantage for systems of n≥2 qubits and are, therefore, fundamental tools for investigating the interface between classical and quantum physics. A theorem by Arkhipov (arXiv:1209.3819) states that n-qubit magic sets in which each observable is in exactly two subsets of compatible observables can be reduced either to the two-qubit magic square or the three-qubit magic pentagram [N. D. Mermin, Phys. Rev. Lett. 65, 3373 (1990)PRLTAO0031-900710.1103/PhysRevLett.65.3373]. An open question is whether there are magic sets that cannot be reduced to the square or the pentagram. If they exist, a second key question is whether they require n>3 qubits, since, if this is the case, these magic sets would capture minimal state-independent quantum advantage that is specific for n-qubit systems with specific values of n. Here, we answer both questions affirmatively. We identify magic sets that cannot be reduced to the square or the pentagram and require n=3, 4, 5, or 6 qubits. In addition, we prove a generalized version of Arkhipov's theorem providing an efficient algorithm for, given a hypergraph, deciding whether or not it can accommodate a magic set, and solve another open problem, namely, given a magic set, obtaining the tight bound of its associated noncontextuality inequality.

Contextuality in entanglement-assisted one-shot classical communication

We consider the problem of entanglement-assisted one-shot classical communication. In the zero-error regime, entanglement can increase the one-shot zero-error capacity of a family of classical channels following the strategy of Cubitt et al., Phys. Rev. Lett. 104, 230503 (2010). This strategy uses the Kochen-Specker theorem which is applicable only to projective measurements. As such, in the regime of noisy states and/or measurements, this strategy cannot increase the capacity. To accommodate generically noisy situations, we examine the one-shot success probability of sending a fixed number of classical messages. We show that preparation contextuality powers the quantum advantage in this task, increasing the one-shot success probability beyond its classical maximum. Our treatment extends beyond Cubitt et al. and includes, for example, the experimentally implemented protocol of Prevedel et al., Phys. Rev. Lett. 106, 110505 (2011). We then show a mapping between this communication task and a corresponding nonlocal game. This mapping generalizes the connection with pseudotelepathy games previously noted in the zero-error case. Finally, after motivating a constraint we term context-independent guessing, we show that contextuality witnessed by noise-robust noncontextuality inequalities obtained in R. Kunjwal, Quantum 4, 219 (2020), is sufficient for enhancing the one-shot success probability. This provides an operational meaning to these inequalities and the associated hypergraph invariant, the weighted max-predictability, introduced in R. Kunjwal, Quantum 3, 184 (2019). Our results show that the task of entanglement-assisted one-shot classical communication provides a fertile ground to study the interplay of the Kochen-Specker theorem, Spekkens contextuality, and Bell nonlocality.

Bosonic fields in states with undefined particle numbers possess detectable non-contextuality features, plus more

The paradoxical features of quantum theory are usually formulated for 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. Is it possible to formulate a contextuality proof for situation in which the numbers of particles are fundamentally undefined? We introduce a representation of the algebra in terms of boson number states in two modes that allows us to assess nonclassicality of states of bosonic fields. This representation allows to show contextuality, and is efficient to reveal violation of local realism, and to formulate entanglement indicators. A form of an non-contextuality inequality is derived, giving a bosonic Peres–Mermin square. The entanglement indicators are built with Pauli-like field observables. The non-clasicality indicators are effective. This is shown for the 2 × 2 bright squeezed vacuum state, and a recently discussed bright-GHZ state resulting from multiple three photon emissions in a parametric process.

Scalable noncontextuality inequalities and certification of multiqubit quantum systems

We propose a family of noncontextuality inequalities and show that they can be used for certification of multiqubit quantum systems. Our scheme, unlike those based on nonlocality, does not require spatial separation between the subsystems, yet it makes use of certain compatibility relations between measurements. Moreover, it is scalable in the sense that the number of expectation values that are to be measured to observe contextuality scales polynomially with the number of qubits that are being certified. In a particular case we also show our scheme to be robust to errors and experimental imperfections. Our results seem promising as far as certification of physical setups is concerned in scenarios where spatial separation between the subsystems cannot be guaranteed.

Stronger Hardy-like proof of quantum contextuality

A Hardy-like proof of quantum contextuality is a compelling way to see the conflict between quantum theory and noncontextual hidden variables (NCHVs), as the latter predict that a particular probability must be zero, while quantum theory predicts a nonzero value. For the existing Hardy-like proofs, the success probability tends to 1/2 when the number of measurement settings n goes to infinity. It means the conflict between the existing Hardy-like proof and NCHV theory is weak, which is not conducive to experimental observation. Here we advance the study of a stronger Hardy-like proof of quantum contextuality, whose success probability is always higher than the previous ones generated from a certain n -cycle graph. Furthermore, the success probability tends to 1 when n goes to infinity. We perform the experimental test of the Hardy-like proof in the simplest case of n = 7 by using a four-dimensional quantum system encoded in the polarization and orbital angular momentum of single photons. The experimental result agrees with the theoretical prediction within experimental errors. In addition, by starting from our Hardy-like proof, one can establish the stronger noncontextuality inequality, for which the quantum-classical ratio is higher with the same n , which provides a new method to construct some optimal noncontextuality inequalities. Our results offer a way for optimizing and enriching exclusivity graphs, helping to explore more abundant quantum properties.

Large violations in Kochen Specker contextuality and their applications

It is of interest to study how contextual quantum mechanics is, in terms of the violation of Kochen Specker state-independent and state-dependent non-contextuality inequalities. We present state-independent non-contextuality inequalities with large violations, in particular, we exploit a connection between Kochen–Specker proofs and pseudo-telepathy games to show KS proofs in Hilbert spaces of dimension d ⩾ 217 with the ratio of quantum value to classical bias being . We study the properties of this KS set and show applications of the large violation. It has been recently shown that Kochen–Specker proofs always consist of substructures of state-dependent contextuality proofs called 01-gadgets. We show a one-to-one connection between 01-gadgets in and Hardy paradoxes for the maximally entangled state in . We use this connection to construct large violation 01-gadgets between arbitrary vectors in , as well as novel Hardy paradoxes for the maximally entangled state in , and give applications of these constructions. As a technical result, we show that the minimum dimension of the faithful orthogonal representation of a graph in is not a graph monotone, a result that may be of independent interest.

Experimental demonstration of the violation of the temporal Peres-Mermin inequality using contextual temporal correlations and noninvasive measurements

We present a generalized quantum scattering circuit which can be used to perform non-invasive quantum measurements, and implement it on NMR qubits. Such a measurement is a key requirement for testing temporal non-contextual inequalities. We use this circuit to experimentally demonstrate the violation of the Peres-Mermin inequality (which is the temporal analog of a Klyachko-Can- Binicioglu-Shumovsky (KCBS) inequality), on a three-qubit NMR quantum information processor. Further, we experimentally demonstrate the violation of a transformed Bell-type inequality (the spatial equivalent of the temporal KCBS inequality) and show that its Tsirelson bound is the same as that for the temporal KCBS inequality. In the temporal KCBS scenario, the contextual bound is strictly lower than the quantum temporal and nonlocal bounds.

Verification of Kochen-Specker-type quantum contextuality with a single photon

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 the 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 states encoded by spin, path, and orbital angular momentum, we experimentally verify several state-dependent noncontextual models of the proofs for the Kochen-Specker theorem by testing an extreme simplest Mermin-like inequality.

Recursive construction of noncontextuality inequalities with arbitrary large violations

Contextuality is an intriguing feature of quantum theory that excludes any noncontextual hidden variable theory of our nature. It is therefore a natural and important pursuit to detect as larger contextuality as possible. In the scenario to reveal contextuality through violation of noncontextuality inequality, the pursuit is to construct such inequality with large violations. In this paper, we present a simple method to extend arbitrary state-(in)dependent noncontextuality inequalities into ones with larger violations. By recursively applying the method, one can obtain noncontextuality inequalities with an arbitrary large violation. Moreover, the method also constructs quantum measurements and states to violate the new inequality.

Dimension-dependent noncontextuality inequalities with large contexts

Quantum theory shows a striking nonclassical feature of contextuality, which is also a crucial resource for quantum information processing. In this paper, we study the dependence of violating noncontextuality inequalities against the dimension of quantum systems. To this end, we consider the restrictions imposed on relations among measurements by limitation of dimensions. Based on the graph-theoretic approach to quantum contextuality, we construct noncontextuality inequalities using multiple mutually orthogonal projectors which are unique to high-dimensional quantum systems. We report two typical classes of dimension-dependent noncontextuality inequalities for one of which a violation implies the dimension of the quantum system is higher than a threshold, and for the other, different amounts of violations correspond to different thresholds of dimension. Our paper is expected to inspire more dimension-dependent noncontextuality inequalities.