TY - JOUR
T1 - Reexamination of a multisetting Bell inequality for qudits
AU - Liang, Yeong Cherng
AU - Lim, Chu Wee
AU - Deng, Dong Ling
PY - 2009/11/25
Y1 - 2009/11/25
N2 - The class of d -setting, d -outcome Bell inequalities proposed by Ji and co-workers are reexamined. For every positive integer d>2, we show that the corresponding nontrivial Bell inequality for probabilities provides the maximum classical winning probability of the Clauser-Horne-Shimony-Holt-like game with d inputs and d outputs. We also demonstrate that the general classical upper bounds given by Ji are underestimated, which invalidates many of the corresponding correlation inequalities presented thereof. We remedy this problem, partially, by providing the actual classical upper bound for d≤13 (including nonprime values of d). We further determine that for prime value d in this range, most of these probability and correlation inequalities are tight, i.e., facet-inducing for the respective classical correlation polytope. Stronger lower and upper bounds on the quantum violation of these inequalities are obtained. In particular, we prove that once the probability inequalities are given, their correlation counterparts given by Ji and co-workers are no longer relevant in terms of detecting the entanglement of a quantum state.
AB - The class of d -setting, d -outcome Bell inequalities proposed by Ji and co-workers are reexamined. For every positive integer d>2, we show that the corresponding nontrivial Bell inequality for probabilities provides the maximum classical winning probability of the Clauser-Horne-Shimony-Holt-like game with d inputs and d outputs. We also demonstrate that the general classical upper bounds given by Ji are underestimated, which invalidates many of the corresponding correlation inequalities presented thereof. We remedy this problem, partially, by providing the actual classical upper bound for d≤13 (including nonprime values of d). We further determine that for prime value d in this range, most of these probability and correlation inequalities are tight, i.e., facet-inducing for the respective classical correlation polytope. Stronger lower and upper bounds on the quantum violation of these inequalities are obtained. In particular, we prove that once the probability inequalities are given, their correlation counterparts given by Ji and co-workers are no longer relevant in terms of detecting the entanglement of a quantum state.
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U2 - 10.1103/PhysRevA.80.052116
DO - 10.1103/PhysRevA.80.052116
M3 - Article
AN - SCOPUS:70849120201
SN - 1050-2947
VL - 80
JO - Physical Review A - Atomic, Molecular, and Optical Physics
JF - Physical Review A - Atomic, Molecular, and Optical Physics
IS - 5
M1 - 052116
ER -