We adopt a continuous model to estimate the Grothendieck constants. An analytical formula to compute the lower bounds of Grothendieck constants has been explicitly derived for arbitrary orders, which improves previous bounds. Moreover, our lower boun
d of the Grothendieck constant of order three gives a refined bound of the threshold value for the nonlocality of the two-qubit Werner states.
The nonlocal properties of arbitrary dimensional bipartite quantum systems are investigated. A complete set of invariants under local unitary transformations is presented. These invariants give rise to both sufficient and necessary conditions for the
equivalence of quantum states under local unitary transformations: two density matrices are locally equivalent if and only if all these invariants have equal values.
