  A well-studied scenario in quantum parameter estimation theory arises when
the parameter to be estimated is imprinted on the initial state by a
Hamiltonian of the form $\theta G$. For such "phase shift Hamiltonians" it has
been shown that one cannot improve the channel quantum Fisher information by
adding ancillas and letting the system interact with them. Here we investigate
the general case, where the Hamiltonian is not necessarily a phase shift, and
show that in this case in general it \emph{is} possible to increase the quantum
channel information and to reach an upper bound. This can be done by adding a
term proportional to the derivative of the Hamiltonian, or by subtracting a
term to the original Hamiltonian. Both methods do not make use of any ancillas
and show therefore that for quantum channel estimation with arbitrary
parameter-dependent Hamiltonian, entanglement with an ancillary system is not
necessary to reach the best possible sensitivity. By adding an operator to the
Hamiltonian we can also modify the time scaling of the channel quantum Fisher
information. We illustrate our techniques with NV-center magnetometry and the
estimation of the direction of a magnetic field in a given plane using a single
spin-1 as probe.
