Quantum Correlations in Successive Spin Measurements (original) (raw)

Will-beset by IN -TECH not be replaced by some classes of HVT, namely local and non-contextual HVT. The most celebrated of this kind of HVT was presented by Bohm in 1952 (Bohm, 1952). Bohm, just prior to developing his HV interpretation, introduced a simplified scenario involving two spin-half particles with correlated spins, rather than two particles with correlated positions and momenta as used by EPR. The EPR-Bohm scenario has the advantage of being experimentally accessible. In 1964 John Bell (Bell, 1964) derived an inequality (which is a statistical result, and is called Bell's inequality BI) using locality and reality assumptions of EPR-Bohm, and showed that the singlet state of two spin-1/2 particles violates this inequality, and hence the contradiction with quantum mechanics. Contemporary versions of the argument are based on the Clauser, Horne, Shimony and Holt (CHSH) inequality(Clauser et al., 1969), rather than the original inequality used by Bell. There is a very good reason for that. While Bell's argument applied only to the singlet state, the CHSH inequality is violated by all pure entangled states (Gisin & Peres, 1992). Early versions of CHSH inequalities involved only two observers, each one having a choice of two (mutually incompatible) experiments. The various outcomes of each experiment were lumped into two sets, arbitrarily called +1 and −1. Possible generalizations involve more than two observers, or more than two alternative experiments for each observer, or more than two distinct outcomes for each experiment. We may consider n-partite systems, each subject to a choice of mv-valued measurements. This gives a total of (mv) n experimentally accessible probabilities. The set of Bell inequalities is then the set of inequalities that bounds this region of probabilities to those accessible with a local hidden variable model. Thus for each value of n, m and v the set of local realistic theories is a polytopes bounded by a finite set of linear Bell inequalities. The CHSH inequalities apply to a situation (n, m, v)=(2, 2, 2). Gisin et al (Gisin & Bechmann-Pasquinucci, 1998) have found a family of Bell inequalities for the case with the number of measurements is arbitrary, i.e. (n, m, v)=(2, m,2). Collins et al (Collins, Gisin, Linden, Massar & Popescu., 2002) and Kaszlikowski et al (Kaszlikowski et al., n.d.) have produced inequalities for arbitrarily high dimensional systems, i.e. (n, m, v)=(2, 2, v). The most complete study of Bell inequalities is for the case (n, m, v)=(n,2,2). n-particle generalizations of the CHSH inequality were first proposed by Mermin (Mermin, 1990), and Belinskii and Klyshko (Belinskii & Klyshko, 1993), and have been extended by Werner and Wolf (Werner & Wolf, 2000), and Zukowski and Brukner (Zuckowski & Brukner, 2002) to give the complete set for two dichotomic observables per site. On the theoretical side, " violation of Bell's inequalities" had become synonymous with "non-classical correlation", i.e., entanglement. One of the first papers in which finer distinctions were made was the construction of states with the property that they satisfy all the usual assumptions leading to the Bell inequalities, but can still not be generated by a purely classical mechanism (are not "separable" in modern terminology) (Werner, 1989). This example pointed out a gap between the obviously entangled states (violating a Bell inequality) and the obviously non-entangled ones, which are merely classical correlated (separable). In 1995 Popescu (Popescu, 1995) (and later (Bennett et al., 1996)) narrowed this gap considerably by showing that after local operations and classical communication one could "distill" entanglement, leading once again to violations, even from states not violating any Bell inequality initially. To summarize this phase: it became clear that violations of Bell inequalities, while still a good indicator for the presence of non-classical correlations by no means capture all kinds of "entanglement". Bell inequalities are statistical predictions about measurements made on two particles, 198 Measurements in Quantum Mechanics