  There has been much interest in developing phase estimation schemes which
beat the so-called Heisenberg limit, i.e., for which the phase resolution
scales better than 1/n, where n is a measure of resources such as the average
photon number or the number of atomic qubits. In particular, a number of
nonlinear schemes have been proposed for which the resolution appears to scale
as 1/n^k or even exp(-n), based on optimising the quantum Cramer-Rao bound.
Such schemes include the use of entangled coherent states. However, it may be
shown that the average root mean square errors of the proposed schemes
(averaged over any prior distribution of phase shifts), cannot beat the
Heisenberg limit, and that simple estimation schemes based on entangled
coherent states cannot scale better than 1/n^{1/4}. This paradox is related to
the role of 'bias' in Cramer-Rao bounds, and is only partially ameliorated via
iterative implementations of the proposed schemes. The results are based on new
information-theoretic bounds for the average information gain and error of any
phase estimation scheme, and generalise to estimates of shifts generated by any
operator having discrete eigenvalues.
