Abstract
The theoretical framework for networked quantum sensing has been developed to a great extent in the past few years, but there are still a number of open questions. Among these, a problem of great significance, both fundamentally and for constructing efficient sensing networks, is that of the role of inter-sensor correlations in the simultaneous estimation of multiple linear functions, where the latter are taken over a collection local parameters and can thus be seen as global properties. In this work we provide a solution to this when each node is a qubit and the state of the network is sensor-symmetric. First we derive a general expression linking the amount of inter-sensor correlations and the geometry of the vectors associated with the functions, such that the asymptotic error is optimal. Using this we show that if the vectors are clustered around two special subspaces, then the optimum is achieved when the correlation strength approaches its extreme values, while there is a monotonic transition between such extremes for any other geometry. Furthermore, we demonstrate that entanglement can be detrimental for estimating non-trivial global properties, and that sometimes it is in fact irrelevant. Finally, we perform a non-asymptotic analysis of these results using a Bayesian approach, finding that the amount of correlations needed to enhance the precision crucially depends on the number of measurement data. Our results will serve as a basis to investigate how to harness correlations in networks of quantum sensors operating both in and out of the asymptotic regime.
Highlights
An important task in quantum information science is to devise protocols for multi-parameter metrology and estimation by exploiting the quantum properties of light and matter
We demonstrate that entanglement can be detrimental for estimating non-trivial global properties, and that sometimes it is irrelevant
We have demonstrated that the strength of the inter-sensor correlations that is useful to estimate a given collection of global properties changes substantially for different amounts of data, i.e., for different values of μ. Since this is the same type of behaviour that we had established for single-parameter schemes in [62], we conjecture that the novel effects associated with a limited number of trials, which here have been uncovered using specific examples, are a general feature of non-asymptotic quantum metrology, and that they are generally present in a wide range of experiments operating in the regime of limited data
Summary
An important task in quantum information science is to devise protocols for multi-parameter metrology and estimation by exploiting the quantum properties of light and matter. Networked scenarios where global properties are relevant provide a natural testbed to identify the potential usefulness of entanglement in a broad range of multi-parameter schemes [32, 37] Within this context, the optimal estimation of a single function f(θ) has been extensively studied [32, 33, 37, 46, 49,50,51,52,53,54,55,56,57,58], and it has been established that one can find entangled states that beat the best separable probe when that function is linear [32, 33]. Since the construction of entangled networks is likely to be difficult in practice, these insights may prove to be crucial in the study and implementation of quantum sensing networks that operate with a realistic amount of data
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