Fake violations of the quantum Bell-parameter bound

Shortcomings of experimental techniques are usually assumed to diminish nonclassical properties of quantum systems. Here it is demonstrated that this standard assumption is not true in general. It is theoretically shown that the inability to resolve different photon numbers in photodetection may pseudo-increase a measured Bell parameter. Under proper conditions one even pseudo-violates the quantum Cirel'son bound of the Bell parameter, the corresponding density operator fails to be positive semi-definite. This paradox can be resolved by appropriate squash models.