Abstract
We consider the problem of estimating multiple analytic functions of a set of local parameters via qubit sensors in a quantum sensor network. To address this problem, we highlight a generalization of the sensor symmetric performance bounds of Rubio et. al. [J. Phys. A: Math. Theor. 53 344001 (2020)] and develop a new optimized sequential protocol for measuring such functions. We compare the performance of both approaches to one another and to local protocols that do not utilize quantum entanglement, emphasizing the geometric significance of the coefficient vectors of the measured functions in determining the best choice of measurement protocol. We show that, in many cases, especially for a large number of sensors, the optimized sequential protocol results in more accurate measurements than the other strategies. In addition, in contrast to the the sensor symmetric approach, the sequential protocol is known to always be explicitly implementable. The sequential protocol is very general and has a wide range of metrological applications.
Highlights
It is well established that entanglement in quantum metrology often facilitates more accurate measurements compared to what is possible with unentangled probes [1,2,3,4,5]
One seeks to optimally measure these local parameters directly by selecting an initial state ρ0 for the sensors, a unitary evolution U by which the local parameters are encoded in the state, and a choice of measurement specified by a positive operator-valued measure (POVM)
While measuring a single analytic function of multiple parameters in this setting is a bona fide multiparameter problem, the fact that one seeks a single quantity makes the problem of finding the information-theoretic optimum for the variance of the desired quantity easier than a more general multiparameter problem; in particular, one can make clever use of rigorous bounds originally derived for the single-parameter case [7,11,12]
Summary
It is well established that entanglement in quantum metrology often facilitates more accurate measurements compared to what is possible with unentangled probes [1,2,3,4,5]. While we conjecture that the optimality of this reduction from analytic to linear functions does generalize to the multifunction case, as we do not claim general optimality of the protocols in this work, the reduction may be freely made without having to prove the veracity of this conjecture Having made this reduction to the problem of measuring multiple linear functions in a quantum sensor network, we can connect to previous works addressing the same problem, subject to various simplifying constraints [8,10,36]. The bound on performance given there is for global protocols and is derived from the quantum Cramér-Rao bound [15,16,40,41] subject to the restriction that one considers only a special set of so-called sensor symmetric states Even within this restriction, beyond the case of d = 2, it is an open question whether the states and measurements (POVMs) required to saturate the derived bound exist for all problems [42]. In addition to presenting this alternative protocol, we lay out how the precise geometric features of a given problem impact the performance of this sequential protocol compared to the signed sensor symmetric approach and the simple local protocol
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