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Most results about quantized detection rely strongly on an assumption of independence among random variables. With this assumption removed, little is known. Thus, in this paper, Bayes-optimal binary quantization for the detection of a shift in mean in a pair of dependent Gaussian random variables is studied. This is arguably the simplest meaningful problem one could consider. If results and rules are to be found, they ought to make themselves plain in this problem. For certain problem parametrizations (meaning the signals and correlation coefficient), optimal quantization is achievable via a single threshold applied to each observation-the same as under independence. In other cases, one observation is best ignored or is quantized with two thresholds; neither behavior is seen under independence. Further, and again in distinction from the case of independence, it is seen that in certain situations, an XOR fusion rule is optimal, and in these cases, the implied decision rule is bizarre. The analysis is extended to the multivariate Gaussian problem.