The mystery of the quantum cakes

As one of the cornerstones of quantum mechanics ~QM!, and thus of modern science, it is a bit unfortunate that the concept of entanglement is so difficult to grasp, even for physicists, and even more difficult to convey to the nonphysicist. Described by Schrödinger as that aspect of quantum mechanics ‘‘that enforces its entire departure from classical lines of thought,’’ 1 entangled states are most often associated with nonlocality ~although there are other ways in which quantum mechanics could be said to be nonlocal!. In particular, it was considering the position/momentum entangled state of two particles that led to the contention by Einstein, Podolsky, and Rosen that quantum mechanics might not be a complete description of nature. Although initially the preference between the ‘‘spook-like action at a distance’’ of QM and a more intuitive ‘‘hidden-variable’’ model ~that preserved local realism! was a philosophical one, in 1964 John Bell showed that the two theories gave different predictions for certain gedanken experiments. The wellknown Bell inequalities set limits on the statistical correlations that could be found between two particles described by any local realistic theory; quantum mechanics was predicted to violate these inequalities. Starting with the work of Clauser et al., the ideal inequalities of Bell were modified to relate to real-world experiments. Since then, there have been very many tests of the inequalities, with the vast majority showing good agreement with QM, and, with the inclusion of some supplementary assumptions to account for low detection efficiency and slow or nonrandom switch settings, have disproved local realistic theories. However, even allowing for the auxillary assumptions necessary to date, the results may still seem somewhat unsatisfying to the non-physicist, for the violation appears only at the statistical level, and then only after a fair amount of logical reasoning. While we in no way dispute the validity of the logic, it has in practice been rather difficult to explain it to the casually interested party, this despite some very nice pedagogical articles published on the topic. More recently, Greenberger, Horne, and Zeilinger have demonstrated the inconsistency of quantum theory and local realism without the need for statistics; however, their arguments require three correlated particles, which is also rather complicated to explain in terms of some sort of classical analog. For this reason, the recent work by Hardy, and others, in which the contradiction of quantum theory and local realism is shown for only two particles without the need for inequalities, has great utility in trying to bring quantum mechanics to a wider audience. And the arguments