  In analogy with the usual state estimation problem, we introduce the problem
of state estimation for a pre- and post-selected ensemble. The problem has
fundamental physical significance since, as argued by Y. Aharonov and
collaborators, pre- and post-selected ensembles are the most basic quantum
ensembles. Two new features are shown to appear: 1) information is flowing to
the measuring device both from the past and from the future; 2)because of the
post-selection, certain measurement outcomes can be forced never to occur. Due
to these features, state estimation in such ensembles is dramatically different
from the case of ordinary, pre-selected only ensembles. Here we develop a
general theoretical framework for studying this problem, and illustrate it
through several examples. We also prove a general theorem showing that
information flowing from the future is related to the complex conjugate
information flowing from the past. We emphasize that all state estimation
problems can be extended to the pre- and post-selected situation. The present
work thus lays the foundations of a much more general theory of quantum state
estimation.
