Universality of the Heisenberg limit for estimates of random phase shifts

摘要

The Heisenberg limit traditionally provides a lower bound on the phase uncertainty scaling as 1/u0002Nu0003, wherenu0002Nu0003 is the mean number of photons in the probe. However, this limit has a number of loopholes which potentiallynmight be exploited to achieve measurements with even greater accuracy. Here we close these loopholes bynproving a completely rigorous form of the Heisenberg limit for the average error over all phase shifts, applicablento any estimate of a completely unknown phase shift. Our result gives a completely general, constraint-free, andnnonasymptotic statement of the Heisenberg limit. It holds for all phase estimation schemes, including multiplenpasses, nonlinear phase shifts, multimode probes, and arbitrary measurements.