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arXiv:1006.0500

Contextuality in Quantum Mechanics: Testing the Klyachko Inequality

Jeffrey Bub, Allen Stairs

Published Jun 2, 2010• 8 citations
arXivHTMLPDF

Abstract

The Klyachko inequality is an inequality for the probabiities of the values of five observables of a spin-1 particle, which is satisfied by any noncontextual assignment of values to this set of observables, but is violated by the probabilities defined by a certain quantum state. We describe an experiment between two entangled spin-1 particles to test contextuality via a related inequality. We point out that a test of contextuality by measurements on a single particle to confirm the Klyachko inequality requires an assumption of non-disturbance by the measuring instrument, which is avoided in the two-particle experiment.

DOI
10.48550/arxiv.1006.0500

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The Klyachko inequality is an inequality for the probabiities of the values of five observables of a spin-1 particle, which is satisfied by any noncontextual assignment of values to this set of observables, but is violated by the probabilities defined by a certain quantum state. We describe an experiment between two entangled spin-1 particles to test contextuality via a related inequality. We point out that a test of contextuality by measurements on a single particle to confirm the Klyachko inequality requires an assumption of non-disturbance by the measuring instrument, which is avoided in the two-particle experiment.
Klyachko, Can, Binicioğlu, and Shumovsky (KCBS) have introduced an inequality which holds if there is a noncontextual hidden variable theory. It is called KCBS inequality. Its violation shows a contradiction between predictions of quantum theory and noncontextual hidden variable theories. Bell-Kochen-Specker (BKS) theorem also shows that there are no noncontextual hidden variable theories in quantum theory by showing that there is no value assignment on a set of all observables. Thus, its approach is different from KCBS inequality. In the present paper, we show a logical relation between KCBS inequality and a value assignment, and examine a consequence of this logical relation.
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 nnn-cycle contextuality inequalities of which n=5n=5n=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 nnn. This makes it one of the simpler contextual inequalities to analyze.
Abstract Contextuality and entanglement are fundamental quantum resources for quantum information processing, and contextuality has been identified as a necessary precondition for entanglement. However, direct implementation of this connection in a realistic experiment remains to be demonstrated. In addition, previous experiments face practical challenges in explicitly satisfying the linear constraints required for contextuality tests. In this work, we present an experimental investigation of this relationship using polarization-entangled photons. First, remote state preparation arising from entanglement allows the required linear constraints to be satisfied structurally. Second, our results provide an operational demonstration of the relationship’s asymmetrical nature. While all four tested Bell states violate a general noncontextuality inequality, confirming they are all contextual resources, only a single Bell state violates the specific entanglement witness mapped from it. This work provides a proof-of-principle experimental implementation of the contextuality-entanglement connection and illustrates how remote state preparation can be used to realize the relevant linear constraints in this setting.
Contextuality is a phenomenon at the heart of the quantum mechanical departure from classical behaviour, and has been recently identified as a resource in quantum computation. Experimental demonstration of contextuality is thus an important goal. The traditional form of contextuality -- as violation of a Kochen-Specker inequality -- requires a quantum system with at least three levels, and the status of the assumption of determinism used in deriving those inequalities has been controversial. By considering `unsharp' observables, Liang, Spekkens and Wiseman (LSW) derived an inequality for generalized noncontextual models that doesn't assume determinism, and applies already to a qubit. We experimentally implement the LSW test using the polarization states of a heralded single photon and three unsharp binary measurements. We violate the LSW inequality by more than 16 standard deviations, thus showing that our results cannot be reproduced by a noncontextual subset of quantum theory.
Abstract Self-testing allows for characterising quantum systems under minimal assumptions. However, existing schemes rely on quantum non-locality and cannot be applied to systems that are not entangled. Here, we introduce a robust method that achieves self-testing of individual systems by taking advantage of contextuality. The scheme is based on the simplest contextuality witness for the simplest contextual quantum system—the Klyachko-Can-Binicioglu-Shumovsky inequality for the qutrit. We establish a lower bound on the fidelity of the state and the measurements as a function of the value of the witness under a pragmatic assumption on the measurements. We apply the method in an experiment on a single trapped 40Ca+ and using randomly chosen measurements and perfect detection efficiency. Using the observed statistics, we obtain the first experimental demonstration of self-testing of a single quantum system with negligible deviations from the assumptions.
In this Bachelor’s thesis the following question is answered: Does the inequality posed in the article Klyachko et al [2008] cover the real part of the Bloch surface of a 3D quantum system when used as in Kochen and Specker [1967]? The Klyachko inequality relies on using five measurements to show contextuality of a subset of states on the real part of the Bloch surface. These can now be used in several configurations as present in the Kochen-Specker contextuality proof, by simply rotating the measurements. We show here that these new inequalities will have subsets of violation that eventually cover the entire real part of the Bloch surface. This can be extended to show that all states of a spin 1 system are noncontextual, so that we have recovered a state-independent contextuality proof by using the Klyachko inequality several times. In the final part, an interpretation of this is given and also some recommendations for further research that should be done in the field.