  The familiar approach to quantum radiation following collapse to a black hole
proceeds via Bogoliubov transformations, and yields probabilities for final
outcomes. In our (complex) approach, we find quantum amplitudes, not just
probabilities, by following Feynman's $+i\epsilon$ prescription. Initial and
final data for Einstein gravity and (say) a massless scalar field are specified
on a pair of asymptotically-flat space-like hypersurfaces $\Sigma_I$ and
$\Sigma_F$; both are diffeomorphic to ${\Bbb R}^3$. Denote by $T$ the (real)
Lorentzian proper-time interval between the surfaces, as measured at spatial
infinity. Then rotate: $T\to{\mid}T{\mid}\exp(-i\theta),0<\theta\leq \pi/2$.
The {\it classical} boundary-value problem is expected to be well-posed on a
region of topology $I\times{\Bbb R}^3$, where $I$ is a closed interval. For a
locally-supersymmetric theory, the quantum amplitude should be dominated by the
semi-classical expression $\exp(iS_{\rm class})$, where $S_{\rm class}$ is the
classical action. One finds the Lorentzian quantum amplitude from the limit
$\theta\to 0_+$. In the usual approach, the only possible such final surfaces
are in the strong-field region shortly before the curvature singularity. In our
approach one can put arbitrary smooth gravitational data on $\Sigma_F$,
provided that it has the correct mass $M$ -- the singularity is by-passed in
the analytic continuation. Here, we consider Bogoliubov transformations and
their possible relation to the probability distribution and density matrix in
the traditional approach. We find that our probability distribution for
configurations of the final scalar field cannot be expressed in terms of the
diagonal elements of some non-trivial density-matrix distribution.
