We develop a general theory to estimate magnetic field gradients in quantum metrology. We consider a system of $N$ particles distributed on a line whose internal degrees of freedom interact with a magnetic field. Usually gradient estimation is based on precise measurements of the magnetic field at two different locations, performed with two independent groups of particles. This approach, however, is sensitive to fluctuations of the off-set field determining the level-splitting of the particles and results in collective dephasing. In this work we use the framework of quantum metrology to assess the maximal accuracy for gradient estimation. For arbitrary positioning of particles, we identify optimal entangled and separable states allowing the estimation of gradients with the maximal accuracy, quantified by the quantum Fisher information. We also analyze the performance of states from the decoherence-free subspace (DFS), which are insensitive to the fluctuations of the magnetic offset field. We find that these states allow to measure a gradient directly, without the necessity of estimating the magnetic offset field. Moreover, we show that DFS states attain a precision for gradient estimation comparable to the optimal entangled states. Finally, for the above classes of states we find simple and feasible measurements saturating the quantum Cram\'er-Rao bound.Comment: 22 pages, 8 figures. See also the related work by I. Apellaniz et al. arXiv: 1703.09056 (2017

We investigate phase and frequency estimation with different measurement strategies under the effect of collective phase noise. First, we consider the standard linear estimation scheme and present an experimentally realisable optimization of the initial probe states by collective rotations. We identify the optimal rotation angle for different measurement times. Second, we show that sub-shot noise sensitivity - up to the Heisenberg limit - can be reached in presence of collective phase noise by using differential interferometry, where one part of the system is used to monitor the noise. For this, not only GHZ states but also symmetric Dicke states are suitable. We investigate the optimal splitting for a general symmetric Dicke state at both inputs and discuss possible experimental realisations of differential interferometry.Comment: 17 pages, 6 figures, v2: small revisions, final versio

Quantum metrology is one of the technologies from which we expect a quantum superiority over classical strategies in the near future. For single-parameter estimation, a clear advantage for global strategies based on entanglement can be proven. However, there exist different answers to the question whether global strategies are better than separable strategies for multi-parameter estimation. The putative inconsistency is based on the investigation of different problem settings as well as different interpretations of the word 'separable' leading to different estimation strategies. In this paper, we discuss these different strategies for the special case of estimating a complete set of linear independent global variables of a sensor network (Eldredge et al 2018 Phys. Rev. A 97, 042337, Proctor et al 2017 arXiv:1702.04271) and outline the advantages and disadvantages of the different strategies.