Experimental violation of a Bell's inequality with ef®cient detection

Local realism is the idea that objects have de®nite properties whether or not they are measured, and that measurements of these properties are not affected by events taking place suf®ciently far away. Einstein, Podolsky and Rosen used these reasonable assumptions to conclude that quantum mechanics is incomplete. Starting in 1965, Bell and others constructed mathematical inequalities whereby experimental tests could distinguish between quantum mechanics and local realistic theories. Many experiments have since been done that are consistent with quantum mechanics and inconsistent with local realism. But these conclusions remain the subject of considerable interest and debate, and experiments are still being re®ned to overcome `loopholes' that might allow a local realistic interpretation. Here we have measured correlations in the classical properties of massive entangled particles (Be ions): these correlations violate a form of Bell's inequality. Our measured value of the appropriate Bell's `signal' is 2:25 6 0:03, whereas a value of 2 is the maximum allowed by local realistic theories of nature. In contrast to previous measurements with massive particles, this violation of Bell's inequality was obtained by use of a complete set of measurements. Moreover, the high detection ef®ciency of our apparatus eliminates the so-called `detection' loophole. Early experiments to test Bell's inequalities were subject to two primary, although seemingly implausible, loopholes. The ®rst might be termed the locality or `lightcone' loophole, in which the correlations of apparently separate events could result from unknown subluminal signals propagating between different regions of the apparatus. Aspect has given a brief history of this issue, starting with the experiments of ref. 8 and highlighting the strict relativistic separation between measurements reported by the Innsbruck group. Similar results have also been reported for the Geneva experiment. The second loophole is usually referred to as the detection loophole. All experiments up to now have had detection ef®ciencies low enough to allow the possibility that the subensemble of detected events agrees with quantum mechanics even though the entire ensemble satis®es Bell's inequalities. Therefore it must be assumed that the detected events represent the entire ensemble; a fair-sampling hypothesis. Several proposals for closing this loophole have been made; we believe the experiment that we report here is the ®rst to do so. Another feature of our experiment is that it uses massive particles. A previous test of Bell's inequality was carried out on protons, but the interpretation of the detected events relied on quantum mechanics, as symmetries valid given quantum mechanics were used to extrapolate the data to a complete set of Bell's angles. Here we do not make such assumptions. A Bell measurement of the type suggested by Clauser, Horne, Shimony and Holt (CHSH) consists of three basic ingredients (Fig. 1a). First is the preparation of a pair of particles in a repeatable starting con®guration (the output of the `magic' box in Fig. 1a). Second, a variable classical manipulation is applied independently to each particle; these manipulations are labelled f1 and f2. Finally, in the detection phase, a classical property with two possible outcomes is measured for each of the particles. The correlation of these outcomes q…f1;f2† ˆ N same…f1;f2†2 Ndifferent…f1;f2† N same ‡ Ndifferent …1†