  The Heisenberg limit traditionally provides a lower bound on the phase
uncertainty scaling as 1/<N>, where <N> is the mean number of photons in the
probe. However, this limit has a number of loopholes which potentially might be
exploited, to achieve measurements with even greater accuracy. Here we close
these loopholes by proving a completely rigorous form of the Heisenberg limit
for the average error over all phase shifts. Our result gives the first
completely general, constraint-free and non-asymptotic statement of the
Heisenberg limit. It holds for all phase estimation schemes, including multiple
passes, nonlinear phase shifts, multimode probes, and arbitrary measurements.
