### Multiparameter Estimation in Networked Quantum Sensors

#### Abstract

We introduce a general model for a network of quantum sensors, and we use this model to consider the following question: When can entanglement between the sensors, and/or global measurements, enhance the precision with which the network can measure a set of unknown parameters? We rigorously answer this question by presenting precise theorems proving that for a broad class of problems there is, at most, a very limited intrinsic advantage to using entangled states or global measurements. Moreover, for many estimation problems separable states and local measurements are optimal, and can achieve the ultimate quantum limit on the estimation uncertainty. This immediately implies that there are broad conditions under which simultaneous estimation of multiple parameters cannot outperform individual, independent estimations. Our results apply to any situation in which spatially localized sensors are unitarily encoded with independent parameters, such as when estimating multiple linear or nonlinear optical phase shifts in quantum imaging, or when mapping out the spatial profile of an unknown magnetic field. We conclude by showing that entangling the sensors can enhance the estimation precision when the parameters of interest are global properties of the entire network.

• Received 19 July 2017
• Corrected 23 February 2018

DOI:https://doi.org/10.1103/PhysRevLett.120.080501

© 2018 American Physical Society

#### Physics Subject Headings (PhySH)

1. Research Areas
Quantum Information

#### Corrections

23 February 2018

Correction: The Acknowledgment section was inadvertently omitted and has now been inserted.

#### Authors & Affiliations

Timothy J. Proctor1,2,*, Paul A. Knott3,4,†, and Jacob A. Dunningham4

• 1Sandia National Laboratories, Livermore, California 94550, USA
• 2Department of Chemistry, University of California, Berkeley, California 94720, USA
• 3Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems (CQNE), School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom
• 4Department of Physics and Astronomy, University of Sussex, Brighton BN1 9QH, United Kingdom

• *tjproct@sandia.gov
• Paul.Knott@nottingham.ac.uk

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##### Issue

Vol. 120, Iss. 8 — 23 February 2018

##### Access Options

Article part of CHORUS

Accepted manuscript will be available starting 21 February 2019.
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#### Images

• ###### Figure 1

A network of quantum sensors. The $k$th node represents a sensor into which the vector parameter ${\mathbit{\varphi }}_{\left[k\right]}$ is encoded via a local unitary evolution. The connections between the nodes denote that, in general, the sensors can be entangled, and/or global measurements can be performed.

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