# Bell's Inequality

In 1935 Einstein, Podolsky and Rosen (EPR) have published their famous paper raising the question whether quantum mechanical description of nature can be considered complete. In particular the question of realism, i.e., that a physical quantity can have a defined value before any observation has taken place, was a point of concern. As quantum mechanics was not able to satisfy their demands of *realism*, *locality* and *completeness*, they conjectured that quantum mechanics is not a complete physical theory. However, by introduction of additional parameters which would prescribe the outcomes of any measurement performed on the system, the theory might be completed (and would also become deterministic). These additional parameters prescribing the outcomes of a measurement, (or more generally their probabilities) are nowadays called local hidden variables (LHV).

Whether such parameters really describe the physical world remained a purely philosophical question until 1964 when John Bell has discovered an inequality, which (with certain modifications) allows assessing this question experimentally. The test can be performed by measuring correlations on pairs of (spin-1/2) particles. Bell's inequality predicts a limit on the degree of correlations for any LHV theory (S≤2 in CHSH formulation), however the predictions of quantum mechanics for certain two-particle states (entangled states) should exceed this limit (up to 2√2).

Since then many experiments testing Bell's inequality were performed. However, there are two major requirements which are necessary for a conclusive test:

- *Detection efficiency*: at least 82.8% of particles have to be detected on each side (under certain conditions this limit can be reduced to 66.7%),

- *Locality*: the events of the measurement (including the choice of the measurement basis) on the two sides must be space-like separated, i.e., causally independent.

An experiment which does not fulfill both requirements can violate Bell's inequality only with additional assumptions, thereby leaving space for LHV theories (open loopholes). Closing both loopholes at the same time is a great experimental challenge and could be achieved only recently.

Our approach is to entangle two remotely trapped atoms by first entangling each of them with a photon and then performing a Bell-state measurement on the photons. The high detection efficiency achievable with atoms together with the possibility to detect the atomic state within a very short time (~1 µs) allows us to perform successful loophole-free tests of Bell's inequality.