  We consider an instance of black-box quantum metrology in the Gaussian
framework, where we aim to estimate the amount of squeezing applied on an input
probe, without previous knowledge on the phase of the applied squeezing. By
taking the quantum Fisher information (QFI) as the figure of merit, we evaluate
its average and variance with respect to this phase in order to identify probe
states that yield good precision for many different squeezing directions. We
first consider the case of single-mode Gaussian probes with the same energy,
and find that pure squeezed states maximize the average quantum Fisher
information (AvQFI) at the cost of a performance that oscillates strongly as
the squeezing direction is changed. Although the variance can be brought to
zero by correlating the probing system with a reference mode, the maximum AvQFI
cannot be increased in the same way. A different scenario opens if one takes
into account the effects of photon losses: coherent states represent the
optimal single-mode choice when losses exceed a certain threshold and,
moreover, correlated probes can now yield larger AvQFI values than all
single-mode states, on top of having zero variance.
