  In interferometry, sub-Heisenberg strategies claim to achieve a phase
estimation error smaller than the inverse of the mean number of photons
employed (Heisenberg bound). Here we show that one can achieve a comparable
precision without performing any measurement, just using the large prior
information that sub-Heisenberg strategies require. For uniform prior (i.e. no
prior information), we prove that these strategies cannot achieve more than a
fixed gain of about 1.73 over Heisenberg-limited interferometry. Analogous
results hold for arbitrary single-mode prior distributions. These results
extend also beyond interferometry: the effective error in estimating any
parameter is lower bounded by a quantity proportional to the inverse
expectation value (above a ground state) of the generator of translations of
the parameter.
