Quantum-mechanical realization of a Popescu-Rohrlich box

We consider quantum ensembles which are determined by pre- and post-selection. Unlike the case of only preselected ensembles, we show that in this case the probabilities for measurement outcomes at intermediate times satisfy causality only rarely; such ensembles can in general be used to signal between causally disconnected regions. We show that under restrictive conditions, there are certain nontrivial bipartite ensembles which do satisfy causality. These ensembles give rise to a violation of the Clauser-Horne- Shimony-Holt inequality, which exceeds the maximal quantum violation given by Tsirelson's bound BCHSH ≤22 and obtains the Popescu-Rohrlich bound for the maximal violation, BCHSH ≤4. This may be regarded as an a posteriori realization of supercorrelations, which have recently been termed Popescu-Rohrlich boxes.