  We consider the problem of estimating the phase of squeezed vacuum states
within a Bayesian framework. We derive bounds on the average Holevo variance
for an arbitrary number $N$ of uncorrelated copies. We find that it scales with
the mean photon number, $n$, as dictated by the Heisenberg limit, i.e., as
$n^{-2}$, only for $N>4$. For $N\leq 4$ this fundamental scaling breaks down
and it becomes $n^{-N/2}$. Thus, a single squeezed vacuum state performs worse
than a single coherent state with the same energy. We find the optimal
splitting of a fixed given energy among various copies. We also compute the
variance for repeated individual measurements (without classical communication
or adaptivity) and find that the standard Heisenberg-limited scaling $n^{-2}$
is recovered for large samples.
