  We discuss the concept of connection states (or connection matrices) that
describe posterior ensembles, post-selected according to the outcomes of a
quantum measurement. Connection matrices allow one to obtain results of any
weak and some non-weak pre- and post-selected measurements, in the same manner
as density matrices allow one to predict the results of conventional quantum
measurements. Connection matrices are direct extensions of the density matrices
and are generally non-Hermitian, which we show to be a direct consequence of
quantum complementarity. This implies that the ultimate reason for unusual weak
values is quantum complementarity. We show that connection matrices can be
determined experimentally. We also show that retrodictive states are a special
case of connection states. We propose a new method of tomography of quantum
detectors.
