Local versus global strategies in multiparameter estimation

We consider the problem of estimating multiple phases using a multimode interferometer. In this setting we show that while global strategies that estimate all the phases simultaneously can lead to high precision gains, the same enhancements can be obtained with local strategies in which each phase is estimated individually. A key resource for the enhancement is shown to be a large particle-number variance in the probe state, and for states where the total particle number is not fixed, this can be obtained for mode-separable states, and the phases can be read out with local measurements. This has important practical implications because local strategies are generally preferred to global ones for their robustness to local estimation failure, flexibility in the distribution of resources, and comparatively easier state preparation. We obtain our results by analyzing two different schemes: the first uses a set of interferometers, which can be used as a model for a network of quantum sensors, and the second looks at measuring a number of phases relative to a reference, which is concerned primarily with quantum imaging.

Figure 1

The general problem under consideration consists of $M$ optical modes with independent linear phase shifts ${\theta }_i, i=1,\cdots ,M$. In optical interferometry the parameters to be estimated, ${\varphi }_i$, are given by some function of the $M$-dimensional vector $\mathbf{\theta }$, as described in the main text. For example, ${\varphi }_i$ could be phase differences between arms.

Figure 2

A network of quantum sensors may be modeled as $d$ parallel interferometers. The parameters to be measured are the phase differences in each interferometer.