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.

Jan 1, 20106 citations

A quantum Bell inequality for infinitely many outcomes

Stefan Zohren, P.M. Reska, Richard David Gill +1 more

Mar 2, 20100 citations

Quantum bounds on Bell inequalities

Károly F. Pál, Tamás Vértesi

We have determined the maximum quantum violation of 241 tight bipartite Bell inequalities with up to five two-outcome measurement settings per party by constructing the appropriate measurement operators in up to six-dimensional complex and eight-dimensional real-component Hilbert spaces using numerical optimization. Out of these inequalities 129 have been introduced here. In 43 cases higher-dimensional component spaces gave larger violation than qubits, and in three occasions the maximum was achieved with six-dimensional spaces. We have also calculated upper bounds on these Bell inequalities using a method proposed recently. For all but 20 inequalities the best solution found matched the upper bound. Surprisingly, the simplest inequality of the set examined, with only three measurement settings per party, was not among them, despite the high dimensionality of the Hilbert space considered. We also computed detection threshold efficiencies for the maximally entangled qubit pair. These could be lowered in several instances if degenerate measurements were also allowed.We have determined the maximum quantum violation of 241 tight bipartite Bell inequalities with up to five two-outcome measurement settings per party by constructing the appropriate measurement operators in up to six-dimensional complex and eight-dimensional real-component Hilbert spaces using numerical optimization. Out of these inequalities 129 have been introduced here. In 43 cases higher-dimensional component spaces gave larger violation than qubits, and in three occasions the maximum was achieved with six-dimensional spaces. We have also calculated upper bounds on these Bell inequalities using a method proposed recently. For all but 20 inequalities the best solution found matched the upper bound. Surprisingly, the simplest inequality of the set examined, with only three measurement settings per party, was not among them, despite the high dimensionality of the Hilbert space considered. We also computed detection threshold efficiencies for the maximally entangled qubit pair. These could be lowered in several instances if degenerate measurements were also allowed.

arXiv:0810.1615Physical Review AFeb 26, 200985 citations

MAXIMAL QUANTUM VIOLATION OF THE CGLMP INEQUALITY ON ITS BOTH SIDES

Ming-Guang Hu, Dong-Ling Deng, Jing‐Ling Chen

We investigate the maximal violations for both sides of the d-dimensional CGLMP inequality by using the Bell operator method. It turns out that the maximal violations have a decelerating increase as the dimension increases and tend to a finite value at infinity. The numerical values are given out up to d = 106 for positively maximal violations and d = 2 × 105 for negatively maximal violations. Counterintuitively, the negatively maximal violations tend to be a little stronger than the positively maximal violations. Further we show the states corresponding to these maximal violations and compare them with the maximally entangled states by utilizing entangled degree defined by von Neumann entropy. It shows that their entangled degree tends to some nonmaximal value as the dimension increases.We investigate the maximal violations for both sides of the d-dimensional CGLMP inequality by using the Bell operator method. It turns out that the maximal violations have a decelerating increase as the dimension increases and tend to a finite value at infinity. The numerical values are given out up to d = 106 for positively maximal violations and d = 2 × 105 for negatively maximal violations. Counterintuitively, the negatively maximal violations tend to be a little stronger than the positively maximal violations. Further we show the states corresponding to these maximal violations and compare them with the maximally entangled states by utilizing entangled degree defined by von Neumann entropy. It shows that their entangled degree tends to some nonmaximal value as the dimension increases.

arXiv:0802.0411International Journal of Quantum InformationOct 1, 20083 citations

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

Stefan Zohren, Richard David Gill

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.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.

arXiv:quant-ph/0612020Physical Review LettersMar 27, 2008113 citations

Violation of left-right symmetry in entanglement space

Ming-Guang Hu, Dong-Ling Deng, Jing‐Ling Chen

We investigate the maximal violations for both sides of the $d$ -dimensional CGLMP inequality by using the Bell operator method. It turns out that the maximal violations have a decelerating increase as the dimension increases and tend to a finite value at infinity. The numerical values are given out up to $d=10^6$ for positively maximal violations and $d=2\times 10^5$ for negatively maximal violations. Counterintuitively, the negatively maximal violations tend to be a little stronger than the positively maximal violations. Further we show the states corresponding to these maximal violations and compare them with the maximally entangled states by utilizing entangled degree defined by von Neumann entropy. It shows that their entangled degree tends to some nonmaximal value as the dimension increases.We investigate the maximal violations for both sides of the $d$-dimensional CGLMP inequality by using the Bell operator method. It turns out that the maximal violations have a decelerating increase as the dimension increases and tend to a finite value at infinity. The numerical values are given out up to $d=10^6$ for positively maximal violations and $d=2\times 10^5$ for negatively maximal violations. Counterintuitively, the negatively maximal violations tend to be a little stronger than the positively maximal violations. Further we show the states corresponding to these maximal violations and compare them with the maximally entangled states by utilizing entangled degree defined by von Neumann entropy. It shows that their entangled degree tends to some nonmaximal value as the dimension increases.

Feb 4, 20080 citations

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.

We investigate the maximal violations for both sides of the

d

-dimensional CGLMP inequality by using the Bell operator method. It turns out that the maximal violations have a decelerating increase as the dimension increases and tend to a finite value at infinity. The numerical values are given out up to

d=10^6

for positively maximal violations and

d=2\times 10^5

for negatively maximal violations. Counterintuitively, the negatively maximal violations tend to be a little stronger than the positively maximal violations. Further we show the states corresponding to these maximal violations and compare them with the maximally entangled states by utilizing entangled degree defined by von Neumann entropy. It shows that their entangled degree tends to some nonmaximal value as the dimension increases.