  Using the space-time analogy, we compare the performance of quantum temporal
imaging with its classical counterpart. We consider a temporal imaging scheme,
based on the sum-frequency generation (SFG) time lens, but our results can be
applied to other temporal imaging schemes such as, for instance, four-wave
mixing. Extending the theory presented in our previous publications, in this
paper we take into account the finite time aperture of the imaging system,
characterized by its pupil function. Using the quantum theory, we obtain a
unitary transformation of the quantum field from the input to the output of the
imaging scheme and identify the contribution of the vacuum fluctuations missing
in the classical theory. This contribution plays a key role in the quantum
temporal imaging of nonclassical temporal waveforms, characterized by
nonclassical fluctuations of the electromagnetic field. As an example, we
consider quantum temporal imaging of broadband squeezed light and formulate the
criteria for conservation of its squeezing properties at the output of the
system.
