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arXiv:quant-ph/0208150 · Physical Review A

Maximal violation of the Clauser-Horne-Shimony-Holt inequality for two qutrits

Libin Fu, Jing‐Ling Chen, Xian-Geng Zhao

Published Aug 28, 2003• 25 citations
arXivHTMLPDF

Abstract

The Bell--Clauser-Horne-Shimony-Holt (BCHSH) inequality (in terms of correlation functions) of two qutrits is studied in detail by employing tritter measurements. A uniform formula for the maximum value of this inequality for tritter measurements is obtained. Based on this formula, we show that nonmaximally entangled states violate the BCHSH inequality more strongly than the maximally entangled one. This result is consistent with what was obtained by Ac\'{\i}n et al. [Phys. Rev. A $65,$ 052325 (2002)] using the Bell--Clauser-Horne inequality (in terms of probabilities).

DOI
10.1103/physreva.68.022323

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The Bell--Clauser-Horne-Shimony-Holt (BCHSH) inequality (in terms of correlation functions) of two qutrits is studied in detail by employing tritter measurements. A uniform formula for the maximum value of this inequality for tritter measurements is obtained. Based on this formula, we show that nonmaximally entangled states violate the BCHSH inequality more strongly than the maximally entangled one. This result is consistent with what was obtained by Ac'{\i}n et al. [Phys. Rev. A 65,65,65, 052325 (2002)] using the Bell--Clauser-Horne inequality (in terms of probabilities).
Clauser-Horne-Shimony-Holt inequality for bipartite systems of four dimensions is studied in detail by employing the unbiased eight-port beam splitters measurements. The uniform formulas for the maximum and minimum values of this inequality for such measurements are obtained. Based on these formulas, we show that an optimal nonmaximally entangled state is about 66%6 more resistant to noise than the maximally entangled one. We also give the optimal state and the optimal angles which are important for experimental realization.
Maximally entangled states should maximally violate the Bell inequality. It is proved that all two-qubit states that maximally violate the Bell-Clauser-Horne-Shimony-Holt inequality are exactly Bell states and the states obtained from them by local transformations. The proof is obtained by using the certain algebraic properties that Pauli's matrices satisfy. The argument is extended to the three-qubit system. Since all states obtained by local transformations of a maximally entangled state are equally valid entangled states, we thus give the characterizations of maximally entangled states in both the two-qubit and three-qubit systems in terms of the Bell inequality.
We introduce a method of investigating qutrit nonclassicality by translating qutrit operators to symmetric two-qubit operators. We show that this procedure sheds light on the discrepancy between maximal qutrit entanglement and maximal nonclassicality of qutrit correlations. Namely, we express Bell operators corresponding to qutrit Bell inequalities in terms of symmetric two-qubit operators and analyze the maximal quantum violation of a given Bell inequality from the qubit perspective. As an example we show that the two-qutrit Collins-Gisin-Linden-Massar-Popescu (CGLMP) Bell inequality can be seen as a combination of Mermin's and Clauser-Horne-Shimony-Holt CHSH) qubit Bell inequalities, and therefore the optimal state violating this combination differs from the one which corresponds to the maximally entangled state of two qutrits. In addition, we discuss the same problem for a three-qutrit inequality. We also demonstrate that the maximal quantum violation of the CGLMP inequality follows from complementarity of correlations.
Imagine two parties, Alice and Bob, who share an entangled quantum state. A well-established result that if Alice performs a two-outcome measurement on the portion of the state in her possession and Bob does likewise, they are able to produce correlations that cannot be reproduced by any classical theory. The allowed classical correlations can be expressed quantitatively by the Bell inequalities. Here we propose new families of Bell inequalities, as a generalization of the Clauser-Horne-Shimony-Holt inequality and show that the maximum violation of these Bell inequalities allowed by quantum theory cannot be attained by a bipartite quantum system having support on a qubit at each site.
In this paper we show a Clauser-Horne (CH) inequality for two three-level quantum systems or qutrits, alternative to the CH inequality given by Kaszlikowski et al. [Phys. Rev. A65, 032118 (2002)]. In contrast to this latter CH inequality, the new one is shown to be equivalent to the Clauser-Horne-Shimony-Holt (CHSH) inequality for two qutrits given by Collins et al. [Phys. Rev. Lett. 88, 040404 (2002)]. Both the CH and CHSH inequalities exhibit the strongest resistance to noise for a nonmaximally entangled state for the case of two von Neumann measurements per site, as first shown by Acin et al. [Phys. Rev. A65, 052325 (2002)]. This equivalence, however, breaks down when one takes into account the less-than-perfect quantum efficiency of detectors. Indeed, for the noiseless case, the threshold quantum efficiency above which there is no local and realistic description of the experiment for the optimal choice of measurements is found to be [Formula: see text] for the CH inequality, whereas it is equal to [Formula: see text] for the CHSH inequality.
The Clauser-Horne-Shimony-Holt inequality (CHSH) is one of the most popular and well-studied witnesses of Bell's theorem, separating classical from quantum correlations. In this work, for every $d\ensuremath{\ge}2$, we present a generalization of the CHSH inequality with the property that maximal violation is achieved uniquely by the maximally entangled state of local dimension ddd. This provides an avenue for device-independent certification of maximally entangled states of arbitrary local dimension.