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arXiv:1207.1212 · Physical Review A

Contextuality of almost all qutrit states can be revealed with nine observables

Paweł Kurzyński, Dagomir Kaszlikowski

Published Oct 31, 2012• 35 citations
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Abstract

We propose a test of quantum contextuality for a single three-level system that uses nine projective measurements. It has the form of an inequality that has to be satisfied by any noncontextual theory. We show that for any qutrit state, except for the maximally mixed state, one can always find a set of nine observables for which the corresponding inequality is violated. This inequality is a test of contextuality that may offer insight into the relation between state-dependent and state-independent contextuality tests present in the literature.

DOI
10.1103/physreva.86.042125

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We propose a test of quantum contextuality for a single three-level system that uses nine projective measurements. It has the form of an inequality that has to be satisfied by any noncontextual theory. We show that for any qutrit state, except for the maximally mixed state, one can always find a set of nine observables for which the corresponding inequality is violated. This inequality is a test of contextuality that may offer insight into the relation between state-dependent and state-independent contextuality tests present in the literature.
The violation of contextuality inequalities shows a strong conflict between the noncontextual hidden variable theory and quantum mechanics. A single qutrit is the simplest indivisible quantum system that can allow a state-dependent contextuality test with five observables and a state-independent contextuality test with 13 observables. For any qutrit states, except for the maximally mixed state, one can always find a set of nine observables for which the corresponding inequality is violated [P. Kurzynski and D. Kaszlikowski, Phys. Rev. A 86, 042125 (2012)]. In this paper, we demonstrate the violation of this inequality using single-photon experiments. Our experiments not only enrich the tests of quantum contextuality, but also deepen the understanding of the physical nature of quantum mechanics.
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 Bell inequality. We show that this connection also applies to other recently introduced noncontextual inequalities.
Quantum contextuality, as proved by Kochen and Specker, and also by Bell, should manifest itself in any state in any system with more than two distinguishable states and recently has been experimentally verified. However, for the simplest system capable of exhibiting contextuality, a qutrit, the quantum contextuality is verified only state dependently in experiment because too many (at least 31) observables are involved in all the known state-independent tests. Here we report an experimentally testable inequality involving only 13 observables that is satisfied by all noncontextual realistic models while being violated by all qutrit states. Thus our inequality facilitates a state-independent test of the quantum contextuality for an indivisible quantum system. We also provide a record-breaking state-independent proof of the Kochen-Specker theorem with 13 directions determined by 26 points on the surface of a magic cube.
We show that, for any system with a number of levels which can be identified with nnn qubits, there is an inequality for the correlations between three compatible dichotomic measurements which must be satisfied by any noncontextual theory, but is violated by any quantum state. Remarkably, the violation grows exponentially with nnn, and the tolerated error per correlation also increases with nnn, showing that state-independent quantum contextuality is experimentally observable in complex systems.
We report a state-independent experimental test of quantum contextuality of a single-photon qutrit. The experimental results demonstrate violations of an inequality originally formulated by Yu and Oh [Phys. Rev. Lett. 108, 030402 (2012)] and further optimized by Cabello et al. [Phys. Rev. A 85, 032108 (2012)]; this inequality is satisfied by all noncontextual hidden variable models and is violated by all qutrit states. Our experiment shows quantum contextuality of this nature in a most fundamental way: A way that is independent of state and unrelated to entanglement.
Contextuality is a foundational phenomenon underlying key differences between quantum theory and classical realistic descriptions of the world. Here we propose an experimental test which is capable of revealing contextuality in all qutrit systems, except the completely mixed state, provided we choose the measurement basis appropriately. The 3-level system is furnished by the polarization and spatial degrees of freedom of a single photon, which encompass three orthogonal modes. Projective measurements along rays in the 3-dimensional Hilbert space are made by linear optical elements and detectors which are sensitive to single mode. We also discuss the impact of detector inefficiency and losses and review the theoretical foundations of this test.