Detection efficiency for loophole-free Bell tests with entangled states affected by colored noise

Loophole-free Bell tests for quantum nonlocality and long-distance secure communication require photodetection efficiencies beyond a threshold eta(crit) that depends on the Bell inequality and the noise affecting the entangled state received by the distant parties. Most calculations of eta(crit) assume that the noise is random and can be modeled as white noise. However, most sources suffer from colored noise. Indeed, since entangled states are usually created as a superposition of two possible deexcitation paths, a partial distinguishability between the two processes leads to the appearance of colored noise in the generated state. Recently, there was a proposal for a loophole-free Bell test [A. Cabello and F. Sciarrino, Phys. Rev. X 2, 021010 (2012)], where a specific colored noise appears as a consequence of the precertification of the photon's presence through single-photon spontaneous parametric down-conversion. Here we obtain eta(crit), the optimal quantum states, and the local settings for a loophole-free Bell test as a function of the amount of colored noise. We consider three bipartite Bell inequalities with n dichotomic settings: Clauser-Horne-Shimony-Holt (n = 2), I-3322 (n = 3), and A(5) (n = 4), both for the case of symmetric efficiencies, corresponding to photon-photon Bell tests, and for the totally asymmetric case, corresponding to atom-photon Bell tests. Remarkably, in all these cases, eta(crit) is robust against the colored noise. The present analysis can find application in any test of Bell inequalities in which the dominant noise is of the colored type. DOI: 10.1103/PhysRevA.87.012113