On the equivalence of the Clauser–Horne and Eberhard inequality based tests

Abstract

Recently, the results of the first experimental test for entangled photons\nclosing the detection loophole (also referred to as the fair sampling loophole)\nwere published (Vienna, 2013). From the theoretical viewpoint the main\ndistinguishing feature of this long-aspired experiment was that the Eberhard\ninequality was used. Almost simultaneously another experiment closing this\nloophole was performed (Urbana-Champaign, 2013) and it was based on the\nClauser-Horne inequality (for probabilities). The aim of this note is to\nanalyze the mathematical and experimental equivalence of tests based on the\nEberhard inequality and various forms on the Clauser-Horne inequality. The\nstructure of the mathematical equivalence is nontrivial. In particular, it is\nnecessary to distinguish between algebraic and statistical equivalence.\nAlthough the tests based on these inequalities are algebraically equivalent,\nthey need not be equivalent statistically, i.e., theoretically the level of\nstatistical significance can drop under transition from one test to another (at\nleast for finite samples). Nevertheless, the data collected in the Vienna-test\nimplies not only a statistically significant violation of the Eberhard\ninequality, but also of the Clauser-Horne inequality (in the ratio-rate form):\nfor both a violation $>60\\sigma.$\n