  Quantum tomography is the standard method of reconstructing the Wigner
function of quantum states of light by means of balanced homodyne detection.
The reconstruction quality strongly depends on the photodetectors quantum
efficiency and other losses in the measurement setup. In this article we
analyse in detail a protocol of enhanced quantum tomography, proposed by
Leonhardt and Paul in 1994, which allows one to reduce the degrading effect of
detection losses. It is based on phase sensitive parametric amplification, with
the phase of the amplified quadrature being scanned synchronously with the
local oscil- lator phase. Although with sufficiently strong amplification the
protocol enables overcoming any detection inefficiency, it was so far not
implemented in experiment, probably due to the losses in the amplifier. Here we
discuss a possible proof-of-principle experiment with a traveling-wave
parametric amplifier. We show that with the state-of-the art optical elements,
the protocol enables high-fidelity tomographic reconstruction of bright
nonclassical states of light. We consider two examples: bright squeezed vacuum
and squeezed single-photon state, with the latter being a non-Gaussian state
and both strongly affected by the losses.
