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arXiv:2009.08734 · AIP conference proceedings

Retrit states violating the KCBS inequality and necessary conditions for maximal contextuality

Fırat Diker, Z. Gedik

Published Jan 1, 2023• 0 citations

Abstract

Since violations of inequalities implied by non-contextual and local hidden variable theories are observed, it is essential to determine the set of (non-)contextual states. Along this direction, one should determine the conditions under which quantum contextuality is observed. It is also important to determine how one can find maximally contextual qutrits. In this work, we revisit the Klyachko-Can-Binicioğlu-Shumovsky (KCBS) scenario where we observe a five-measurement state-dependent contextuality. We investigate possible symmetries of the KCBS pentagram, i.e., the conservation of the contextual characteristic of a qutrit-system. For this purpose, the KCBS operator including five cyclic measurements is rotated around the Z-axis. We then check a set of rotation angles to determine the contextuality and non-contextuality regions for the eigenstates of the spin-1 operator for an arbitrary rotation. We perform the same operation for the homogeneous linear combination of the eigenstates with spin values +1 and -1. More generally, we work on the real subgroup of the three dimensional Hilbert space to determine the set of (non-)contextual states under certain rotations in the physical Euclidean space $\mathbb{E}^3$. Finally, we show data on Euler rotation angles for which maximally contextual retrits (qutrits of the real Hilbert space) are found, and derive mathematical relations through data analysis between Euler angles and qutrit states parameterized with spherical coordinates.

DOI: 10.1063/5.0163019

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\mathbb{E}^3

. Finally, we show data on Euler rotation angles for which maximally contextual retrits (qutrits of the real Hilbert space) are found, and derive mathematical relations through data analysis between Euler angles and qutrit states parameterized with spherical coordinates.

Klyachko–Can–Binicioglu–Shumovsky (KCBS) inequality is a Bell-like inequality,the violation of which can be used to confirm the existence of quantum contextuality.However,the imperfection of detection efficiency may cause the so-called loophole in actual KCBS’s experiments.We derive an alternative KCBS inequality to deal with the loophole in actual KCBS’s experiments.We prove that if the experimental data violate this KCBS inequality,the loophole-free violation of the original KCBS inequality will occur.We show that the minimum detection efficiency needed for a loophole-free violation of the KCBS inequality is about 0.9738.

The Quad-

C_5

Graph: Maximum Contextuality Gap on Eight Vertices

Quantum measurements can exhibit contextuality: their outcomes cannot always be explained by assigning pre-existing values that are independent of which compatible measurements are performed together. The Klyachko-Can-Binicioğlu-Shumovsky (KCBS) inequality provides the canonical minimal test of this effect for a single three-level quantum system, or qutrit, using five measurement events arranged as a pentagon. Here we ask whether a larger but still compact set of measurement events can produce a stronger separation between quantum predictions and the corresponding noncontextual limit. We perform an exhaustive search over all 11,117 connected non-isomorphic graphs with eight vertices, where each vertex represents a measurement event and edges connect pairs of events that cannot occur together. We identify a sparse ten-edge graph, which we call Quad-

C_5

, as the unique maximizer of this separation at the reported numerical precision. The graph can be understood as four overlapping KCBS pentagons, with every edge shared by two pentagons. Quad-

C_5

already demonstrates contextuality in a qutrit, for which we obtain an exact analytical result and find the same violation above the noncontextual bound as in the original KCBS test. The larger quantum-noncontextual separation allowed by the graph, however, becomes accessible in a four-level quantum system, where numerical optimization reaches the full graph-theoretic quantum bound and yields a larger contextuality gap than the standard eight-vertex Wagner-graph benchmark while requiring fewer pairwise constraints. Quad-

C_5

therefore provides a compact connection between minimal qutrit contextuality and stronger contextuality tests available in higher-dimensional quantum systems.

We present a study of quantum contextuality of three-dimensional mixed states for the Klyachko-Can-Binicio\ifmmode \breve{g}\else \u{g}\fi{}lu-Shumovsky (KCBS) and the Kurzy\ifmmode \acute{n}\else \'{n}\fi{}ski-Kaszlikowski (KK) noncontextuality inequalities. For any class of states whose eigenvalues are arranged in decreasing order, a universal set of measurements always exists for the KK inequality, whereas none exists for the KCBS inequality. This difference can be reflected from the spectral distribution of the overall measurement matrix. Our results would facilitate the error analysis for experimental setups, and our spectral method in the paper combined with graph theory could be useful in future studies on quantum contextuality.

Klyachko—Can—Binicioğlu—Shumovsky (KCBS) inequality is a Bell-like inequality, the violation of which can be used to confirm the existence of quantum contextuality. However, the imperfection of detection efficiency may cause the so-called loophole in actual KCBS's experiments. We derive an alternative KCBS inequality to deal with the loophole in actual KCBS's experiments. We prove that if the experimental data violate this KCBS inequality, the loophole-free violation of the original KCBS inequality will occur. We show that the minimum detection efficiency needed for a loophole-free violation of the KCBS inequality is about 0.9738.

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We investigate the set of qutrit states in terms of symmetric states of two qubits that violate the minimal contextual inequality, namely the Klyachko-Can-Binicoglu-Shumovsky (KCBS) inequality. The physical system that provides a natural framework for this problem is a biphoton, which consists of two photons in the same spatiotemporal mode and whose effective polarization behaves as a three-level quantum system. The relationship between the KCBS contextual inequality and the Clauser-Horne-Shimony-Holt (CHSH) inequality is investigated. We find that every biphotonic state that is contextual with respect to KCBS is nonlocal as per the CHSH test when the two photons are apart, but the converse is not true.