Optimized entanglement for quantum parameter estimation from noisy qubits
François Chapeau‐Blondeau · International Journal of Quantum Information · 2018
For parameter estimation from an [Formula: see text]-component composite quantum system, it is known that a separable preparation leads to a mean-squared estimation error scaling as [Formula: see text] while an entangled preparation can in some conditions afford a smaller error with [Formula: see text] 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 [Formula: see text]. To complement this asymptotic characterization, here we characterize how the estimation efficiency evolves as a function of the size [Formula: see text] 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 [Formula: see text] the efficiency of the optimal separable preparation to that of an entangled preparation with arbitrary degree of entanglement. We exhibit the [Formula: see text] superefficiency with no noise, and prove its asymptotic disappearance at large [Formula: see text] for any nonvanishing noise configuration. For maximizing the estimation efficiency, we characterize the existence of an optimum [Formula: see text] 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 [Formula: see text] qubits into independent blocks formed of [Formula: see text] entangled qubits restores at large [Formula: see text] a nonvanishing efficiency that can improve over that of [Formula: see text] independent qubits optimally prepared. Also, one inactive qubit included in the entangled probe sometimes stands as the most efficient setting for estimation. The results further attest with new characterizations the subtlety of entanglement for quantum information in the presence of noise, showing that when entanglement is beneficial, maximum efficiency is not necessarily obtained by maximum entanglement but instead by a controlled degree and finite optimal amount of it.