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arXiv:1504.06942 · Physical Review A

Quantum contextuality of a qutrit state

Zhen‐Peng Xu, Hong-Yi Su, Jing‐Ling Chen

Published Jul 6, 2015• 6 citations
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Abstract

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.

DOI
10.1103/physreva.92.012104

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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.
Quantum mechanics exhibits a very peculiar form of contextuality. Identifying and connecting the simplest scenarios in which more general theories can or cannot be more contextual than quantum mechanics is a fundamental step in the quest for the principle that singles out quantum contextuality. The former scenario corresponds to the Klyachko-Can-Binicio\ifmmode \breve{g}\else \u{g}\fi{}lu-Shumovsky (KCBS) inequality. Here we show that there is a simple tight inequality, twin to the KCBS, for which quantum contextuality cannot be outperformed. In a sense, this twin inequality is the simplest tool for recognizing fully contextual quantum correlations.
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.
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 E3\mathbb{E}^3E3. 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.
In a recent paper [arXiv:1301.2887] Ahrens et al. claim that our "(...) experiment on qutrits does not test Klyachko et al.'s inequality, but an inequality with extra correlations" and that the "(...) experiment cannot be considered a proper test of a noncontextuality inequality, since the same observable is measured with different setups in different contexts". We disagree with these claims. In this note we briefly re-state how our test of the non-contextuality inequality was constructed. We explain how we keep the context of measurements when switching between different terms of the tested inequality, and we argue that we did in fact test the Klyachko, Can, Binicioglu and Shumovsky inequality. In doing so, we also clarify why our experiment is indeed a proper test of noncontextual realism.
The Quad-C5C_5C5​ 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-C5C_5C5​, 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-C5C_5C5​ 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-C5C_5C5​ therefore provides a compact connection between minimal qutrit contextuality and stronger contextuality tests available in higher-dimensional quantum systems.
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.