  The phase uncertainty of an unseeded nonlinear interferometer, where the
output of one nonlinear crystal is transmitted to the input of a second crystal
that analyzes it, is commonly said to be below the shot-noise level but highly
dependent on detection and internal loss. Unbalancing the gains of the first
(source) and second (analyzer) crystals leads to a configuration that is
tolerant against detection loss. However, in terms of sensitivity, there is no
advantage in choosing a stronger analyzer over a stronger source, and hence the
comparison to a shot-noise level is not straightforward. Internal loss breaks
this symmetry and shows that it is crucial whether the source or analyzer is
dominating. Based on these results, claiming a Heisenberg scaling of the
sensitivity is more subtle than in a balanced setup.
