Energy and time entangled photons at a wavelength of 1310 nm are produced by parametric downconversion in a KNbO3 crystal and are sent into all-fiber interferometers using a telecom fiber network. The two interferometers of this Franson-type test of the Bell-inequality are located 10.9 km apart from one another. Two-photon fringe visibilities of up to 81.6 % are obtained. These strong nonlocal correlations support the nonlocal predictions of quantum mechanics and provide evidence that entanglement between photons can be maintained over long distances.Comment: 5 pages, REVTeX, 3 postscript figures include

A local hidden variable model exploiting the detection loophole to reproduce exactly the quantum correlation of the singlet state is presented. The model is shown to be compatible with both the CHSH and the CH Bell inequalities. Moreover, it bears the same rotational symmetry as spins. The reason why the model can reproduce the quantum correlation without violating the Bell theorem is that in the model the efficiency of the detectors depends on the local hidden variable. On average the detector efficiency is limited to 75%.

We derive both numerically and analytically Bell inequalities and quantum measurements that present enhanced resistance to detector inefficiency. In particular, we describe several Bell inequalities which appear to be optimal with respect to inefficient detectors for small dimensionality dϭ2,3,4 and two or more measurement settings at each side. We also generalize the family of Bell inequalities described by Collins et al. ͓Phys. Rev. Lett. 88, 040404 ͑2002͔͒ to take into account the inefficiency of detectors. In addition, we consider the possibility for pairs of entangled particles to be produced with probability less than 1. We show that when the pair production probability is small, one should in general use different Bell inequalities than when the pair production probability is high.

to the preexisting fiber birefringence. The polarimetric method for group delay ripple measurements does not suffer any resolution limitation, except of course the spectral resolution of the laser source, which is at least two orders of magnitude smaller than the resolution of the modulation phase shift method.

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