Complementary Detector and State Preparation Error and Classicality in the Spin-j Einstein–Podolsky–Rosen–Bohm Experiment

The spin-j Einstein–Podolsky–Rosen–Bohm experiment is examined in the context of how the quantum theoretic probability distributions for the spin measurement outcomes are to be coarse-grained in order to yield classical behavior in the \(j \rightarrow \infty \) limit. A coarse-graining protocol is found that can be viewed as imperfection either in the detection process or in state preparation process, and is in both viewpoints minimal in the sense that it is no more than what is needed to wash out the execess quantum correlations. In the first point of view the coarse-grained distribution can be written in terms of a Bell-type factorizable hidden variable model wherein the conditional distributions for the spin measurement outcome of each particle is not just nonnegative but actually attains the value zero for some choice of measurement axis. In the second point of view the coarse-grained distribution arises from a spin state whose Wigner function is not just nonnegative but actually attains the value zero for some spin orientations. That such a remarkable dual interpretation should be possible suggests that this type of complementary coarse graining is an intrinsic aspect of how classicality is obtained in the large j limit, but this conclusion remains speculative.