  We consider quantum ensembles which are determined by pre- and
post-selection. Unlike the case of only pre-selected 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 non-trivial bi-partite ensembles which do satisfy
causality. These ensembles give rise to a violation of the CHSH inequality,
which exceeds the maximal quantum violation given by Tsirelson's bound, $B_{\rm
CHSH}\le 2\sqrt2$, and obtains the Popescu-Rohrlich bound for the maximal
violation, $B_{\rm CHSH}\le 4$. This may be regarded as an a posteriori
realization of super-correlations, which have recently been termed
Popescu-Rohrlich boxes.
