  For parameter estimation from an $N$-component composite quantum system, it
is known that a separable preparation leads to a mean-squared estimation error
scaling as $1/N$ while an entangled preparation can in some conditions afford a
smaller error with $1/N^2$ scaling. This quantum superefficiency is however
very fragile to noise or decoherence, and typically disappears with any small
amount of random noise asymptotically at large $N$. To complement this
asymptotic characterization, we characterize how the estimation efficiency
evolves as a function of the size $N$ of the entangled system and its degree of
entanglement. We address a generic situation of qubit phase estimation, also
meaningful for frequency estimation. Decoherence is represented by the broad
class of noises commuting with the phase rotation, which includes depolarizing,
phase-flip, and thermal quantum noises. In these general conditions, explicit
expressions are derived for the quantum Fisher information quantifying the
ultimate achievable efficiency for estimation. We confront at any size $N$ the
efficiency of the optimal separable preparation to that of an entangled
preparation with arbitrary degree of entanglement. We exhibit the $1/N^2$
superefficiency with no noise, and prove its asymptotic disappearance at large
$N$ for any non-vanishing noise configuration. For maximizing the estimation
efficiency, we characterize the existence of an optimum $N_{\rm opt}$ of the
size of the entangled system along with an optimal degree of entanglement. For
nonunital noises, maximum efficiency is usually obtained at partial
entanglement. Grouping the $N$ qubits into independent blocks formed of $N_{\rm
opt}$ entangled qubits restores at large $N$ a nonvanishing efficiency that can
improve over that of $N$ independent qubits optimally prepared. One inactive
qubit in the entangled probe sometimes is most efficient for estimation.

Quantum Physics