  Imagine two parties, Alice and Bob who share an entangled quantum state. A
well-established result that if Alice performs 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 (CHSH) inequality and show
that the maximum violation of these Bell inequalities allowed by quantum theory
can not be attained by a bipartite quantum system having support on a qubit at
each site.
