  We study very generally to what extent the uncertainty with which a phase
shift can be estimated in quantum metrology can be reduced by extending the
Hamiltonian that generates the phase shift to an ancilla system with a Hilbert
space of arbitrary dimension, and allowing arbitrary interactions between the
original system and the ancilla. Such Hamiltonian extensions provide a general
framework for open quantum systems, as well as for "non-linear metrology
schemes" that have been investigated over the last few years. We prove that
such Hamiltonian extensions cannot improve the sensitivity of the phase shift
measurement when considering the quantum Fisher information optimized over
input states.
