# Optimized entanglement for quantum parameter estimation from noisy qubits

## TL;DR

This paper analyzes how entanglement and noise affect quantum parameter estimation efficiency in noisy qubit systems, deriving explicit formulas and identifying optimal system sizes and entanglement degrees for best performance.

## Contribution

It provides a detailed characterization of the evolution of estimation efficiency with system size and entanglement degree under various noise models, including explicit expressions for quantum Fisher information.

## Key findings

- Maximum efficiency at no noise scales as 1/N^2
- Efficiency asymptotically vanishes with any noise at large N
- Optimal entanglement and system size depend on noise type and level

## Abstract

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.