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arXiv:quant-ph/0612020 · Physical Review Letters

Maximal Violation of the Collins-Gisin-Linden-Massar-Popescu Inequality for Infinite Dimensional States

Stefan Zohren, Richard David Gill

Published Mar 27, 2008• 113 citations
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Abstract

We present a much simplified version of the Collins-Gisin-Linden-Massar-Popescu inequality for the $2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}d$ Bell scenario. Numerical maximization of the violation of this inequality over all states and measurements suggests that the optimal state is far from maximally entangled, while the best measurements are the same as conjectured best measurements for the maximally entangled state. For very large values of $d$ the inequality seems to reach its minimal value given by the probability constraints. This gives numerical evidence for a tight quantum Bell inequality (or generalized Csirelson inequality) for the $2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ scenario.

DOI
10.1103/physrevlett.100.120406

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We present a much simplified version of the Collins-Gisin-Linden-Massar-Popescu inequality for the 2\ifmmode×\else\texttimes\fi2\ifmmode×\else\texttimes\fid2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}d2\ifmmode×\else\texttimes\fi2\ifmmode×\else\texttimes\fid Bell scenario. Numerical maximization of the violation of this inequality over all states and measurements suggests that the optimal state is far from maximally entangled, while the best measurements are the same as conjectured best measurements for the maximally entangled state. For very large values of ddd the inequality seems to reach its minimal value given by the probability constraints. This gives numerical evidence for a tight quantum Bell inequality (or generalized Csirelson inequality) for the 2\ifmmode×\else\texttimes\fi2\ifmmode×\else\texttimes\fi\ensuremath∞2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}2\ifmmode×\else\texttimes\fi2\ifmmode×\else\texttimes\fi\ensuremath∞ scenario.
In this Rapid Communication, we show analytically that all pure entangled states of two ddd-dimensional systems (qudits) violate the Collins-Gisin-Linden-Masser-Popescu (CGLMP) inequality. This property was pointed out by Gisin in the qubit case and then generalized by Gisin-Peres and Popescu-Rohrlich to the qudit case based on the Clauser-Horne-Shimony-Holt (CHSH) inequality. We report the Gisin's theorem for two qudits by making use of the CGLMP inequality.
We propose a generalized Bell inequality for two three-dimensional systems with three settings in each local measurement. It is shown that this inequality is maximally violated if local measurements are configured to be mutually unbiased and a composite state is maximally entangled. This feature is similar to Clauser-Horne-Shimony-Holt inequality for two qubits but is in contrast with the two types of inequalities, Collins-Gisin-Linden-Massar-Popescu and Son-Lee-Kim, for high-dimensional systems. The generalization to aribitrary prime-dimensional systems is discussed.
We present a novel tight bound on the quantum violations of the CGLMP inequality in the case of infinitely many outcomes. Like in the case of Tsirelson's inequality the proof of our new inequality does not require any assumptions on the dimension of the Hilbert space or kinds of operators involved. However, it is seen that the maximal violation is obtained by the conjectured best measurements and a pure, but not maximally entangled, state. We give an approximate state which, in the limit where the number of outcomes tends to infinity, goes to the optimal state for this setting. This state might be potentially relevant for experimental verifications of Bell inequalities through multi-dimenisonal entangled photon pairs.
We present a novel tight bound on the quantum violations of the CGLMP inequality in the case of infinitely many outcomes. Like in the case of Tsirelson's inequality the proof of our new inequality does not require any assumptions on the dimension of the Hilbert space or kinds of operators involved. However, it is seen that the maximal violation is obtained by the conjectured best measurements and a pure, but not maximally entangled, state. We give an approximate state which, in the limit where the number of outcomes tends to infinity, goes to the optimal state for this setting. This state might be potentially relevant for experimental verifications of Bell inequalities through multi-dimenisonal entangled photon pairs.
We present a novel tight bound on the quantum violations of the CGLMP inequality in the case of infinitely many outcomes. Like in the case of Tsirelson's inequality the proof of our new inequality does not require any assumptions on the dimension of the Hilbert space or kinds of operators involved. However, it is seen that the maximal violation is obtained by the conjectured best measurements and a pure, but not maximally entangled, state. We give an approximate state which, in the limit where the number of outcomes tends to infinity, goes to the optimal state for this setting. This state might be potentially relevant for experimental verifications of Bell inequalities through multi-dimenisonal entangled photon pairs.
High-dimensional quantum entanglement is drawing attention because it enables us to perform quantum information tasks that are robust against noises. To test the nonlocality of entangled qudits, the Collins-Gisin-Linden-Massar-Popescu (CGLMP) inequality has been proposed and demonstrated using qudits based on orbital angular momentum, time-energy uncertainty, and frequency bins. Here, we report the generation and observation of time-bin entangled ququarts. We implemented a measurement for the CGLMP inequality test using cascaded delay Mach-Zehnder interferometers fabricated by using planar light-wave circuit technology, with which we achieved a precise and stable measurement for time-bin-entangled ququarts. In addition, we generated an optimized entangled state by modulating the pump pulse intensities, with which we can observe the theoretical maximum violation for the CGLMP inequality test. As a result, we successfully observed a Bell-type parameter ${S}_{4}=2.774\ifmmode\pm\else\textpm\fi{}0.025$ violating the CGLMP inequality for the maximally entangled state and an enhanced Bell-type parameter ${S}_{4}=2.913\ifmmode\pm\else\textpm\fi{}0.023$ for the optimized entangled state.