  It is well known that the violation of Bell's inequality in the form given by
Clauser, Horne, Shimony, and Holt (CHSH) in two-qubit systems requires
entanglement, but not vice versa, i.e., there are entangled states which do not
violate the CHSH inequality. Here we compare some standard entanglement
measures with violations of the CHSH inequality (as given by the Horodecki
measure) for two-qubit states generated by Monte Carlo simulations. We describe
states that have extremal entanglement according to the negativity,
concurrence, and relative entropy of entanglement for a given value of the CHSH
violation. We explicitly find these extremal states by applying the generalized
method of Lagrange multipliers based on the Karush-Kuhn-Tucker conditions. The
found minimal and maximal states define the range of entanglement accessible
for any two-qubit states that violate the CHSH inequality by the same amount.
We also find extremal states for the concurrence versus negativity by
considering only such states which do not violate the CHSH inequality.
Furthermore, we describe an experimentally efficient linear-optical method to
determine the highest Horodecki degree of the CHSH violation for arbitrary
mixed states of two polarization qubits. By assuming to have access
simultaneously to two copies of the states, our method requires only six
discrete measurement settings instead of nine settings, which are usually
considered.
