  Current advances in nonlinear optics have made it possible to perform a
homodyne-like tomography of an unknown state without highly efficient detectors
or a strong local oscillator. Thereby, a new experimental direction has been
opened into multimode and large-bandwidth quantum optics. An optical parametric
amplifier (OPA) allows us to reconstruct the quadrature distribution of an
unknown state directly from the measured intensity distribution with high
precision. We propose adding a controllable displacement to the standard
scheme, obtaining an improved method applicable even to asymmetric and
non-Gaussian states while significantly increasing estimation accuracy and
lowering the OPA amplification requirement. To demonstrate the power of our
method, we accurately detect the sub-Planck phase-space structure by a
distillable squeezing from the OPA estimates of various non-Gaussian states.
With the improvements, OPA tomography became a generally applicable
loss-tolerant and efficient alternative to OPA-assisted homodyne detection.
