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For parameter estimation from an $N$-component composite quantum system, itis known that a separable preparation leads to a mean-squared estimation errorscaling as $1/N$ while an entangled preparation can in some conditions afford asmaller error with $1/N^2$ scaling. This quantum superefficiency is howeververy fragile to noise or decoherence, and typically disappears with any smallamount of random noise asymptotically at large $N$. To complement thisasymptotic characterization, we characterize how the estimation efficiencyevolves as a function of the size $N$ of the entangled system and its degree ofentanglement. We address a generic situation of qubit phase estimation, alsomeaningful for frequency estimation. Decoherence is represented by the broadclass of noises commuting with the phase rotation, which includes depolarizing,phase-flip, and thermal quantum noises. In these general conditions, explicitexpressions are derived for the quantum Fisher information quantifying theultimate achievable efficiency for estimation. We confront at any size $N$ theefficiency of the optimal separable preparation to that of an entangledpreparation 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 estimationefficiency, we characterize the existence of an optimum $N_{\rm opt}$ of thesize of the entangled system along with an optimal degree of entanglement. Fornonunital noises, maximum efficiency is usually obtained at partialentanglement. Grouping the $N$ qubits into independent blocks formed of $N_{\rmopt}$ entangled qubits restores at large $N$ a nonvanishing efficiency that canimprove over that of $N$ independent qubits optimally prepared. One inactivequbit in the entangled probe sometimes is most efficient for estimation.

Optimized entanglement for quantum parameter estimation from noisy qubits