  Quantum resources, such as entanglement, can decrease the uncertainty of a
parameter-estimation procedure beyond what is classically possible. This
phenomenon is well described for noiseless systems with asymptotically many
measurement resources by the Quantum Cramer-Rao Bound, but no general
description exists for the regime of limited measurement resources. We address
this problem by defining a Bayesian quantifier for uncertainty suitable for the
regime of limited resources, and by developing a mathematical description for a
parameter-estimation procedure which uses qubit probes to estimate a rotation
angle induced on them. We simulate the qubit estimation scheme in the regime of
limited resources using a single class of probe states. We find that, in
noiseless systems, entanglement between qubits always decreases the uncertainty
of the estimation; however, the quantum advantage decreases as fewer qubits are
used in the estimation. We also find that the presence of strong dephasing
noise removes the quantum advantage completely, regardless of the number of
qubit probes used.
