We analyze conditions for violation of the Bell inequality in the Clauser-Horne-Shimony-Holt form, focusing on Josephson phase qubits. We start the analysis with maximum violation in the ideal case, and then take into account the effects of local measurement errors and decoherence. Special attention is paid to configurations of the qubit measurement directions in the pseudospin space lying within either horizontal or vertical planes; these configurations are optimal in certain cases. Besides local measurement errors and decoherence, we also discuss the effect of measurement crosstalk, which affects both the classical inequality and the quantum result. In particular, we propose a version of the BI which is insensitive to the crosstalk.
We investigate the Bell inequalities derived from the graph states with violations detectable even with the presence of noises, which generalizes the idea of error-correcting Bell inequalities [Phys. Rev. Lett. 101, 080501 (2008)]. Firstly we construct a family of valid Bell inequalities tolerating arbitrary
t-qubit errors involving
3(t+1) qubits, e.g., 6 qubits suffice to tolerate single qubit errors. Secondly we construct also a single-error-tolerating Bell inequality with a violation that increases exponentially with the number of qubits. Exhaustive computer search for optimal error-tolerating Bell inequalities based on graph states on no more than 10 qubits shows that our constructions are optimal for single- and double-error tolerance.