A ug 1 99 9 Bell theorem involving all possible local measurements

The Bell theorem for a pair of two-state systems in a singlet state is formulated for the entire range of measurement settings. Typeset using REVTEX 1 The Bell theorem is usually formulated with the help of the Clauser-Horne [1] or the CHSH inequality [2]. These inequalities are satisfied by any local realistic theory and are violated by quantum mechanical predictions. They involve two apparatus settings at each of the two sides of the experiment. However, generalisation to more than two settings at each side are possible [3], [4], [5], [6]. There are several motivations for such generalisations. First of all new Bell inequalities may be more appropriate in some experimental situations, e.g., the chained Bell inequalities can reveal violation of local realism for the Franson type experiment [7]. Also, the academic question, why only two settings at each side, is that always necessary, is interesting in itself. Further, many of the currently performed quantum interferometric Bell tests did not involve stabilisation of the interferometers at specified settings optimal for the standard Bell inequalities, but rather involved sample scans of the entire interferometric patterns. Thus it is useful to have inequalities that are directly applicable to such data. Here we present a Bell-type inequality that involves all possible settings of the local measuring apparatus for a pair of two-state systems, which is always equivalent to two spin 1 2 particles. The method applied is a development of the one given in [5]. However, here we do not restrict ourselves to pairs of coplanar settings (in the meaning appropriate for two Stern-Gerlach apparatuses). Our method has two characteristic traits. The first one is that it indeed involves the entire range of the measurement parameters. By this, e.g., it distinguishes itself from the limits of infinitely many settings at each side of the so-called chained inequalities [4], in which not every pair of possible settings is utilised. The second one is that the method involves the quantum prediction from the very beginning. As we shall see the quantum prediction determines the structure of our Bell inequality. In a standard Bell-type experiment one has a source emitting two particles, each of which propagates towards one of two spatially separated measuring devices. The particles are described by the maximally entangled state, e.g.,