Anonymous quantum sensing robust against state preparation errors

Networked quantum sensors have several applications such as the mapping of magnetic fields. When the magnetic fields are biomagnetic ones, i.e., they contain some private information, the information of from who non-zero magnetic fields occur has to be protected from eavesdroppers. Anonymous quantum sensing keeps it secret by estimating amplitudes of the magnetic fields without disclosing the positions of non-zero magnetic fields. In this paper, we propose an anonymous quantum sensing protocol that is robust against any independent noise in state preparations. To this end, we devise a quantum state verification protocol for a superposition of Greenberger-Horne-Zeilinger and Dicke states and combine it with the original protocol of anonymous quantum sensing. Our verification protocol can decide whether the fidelity between the ideal and actual states is high or low more efficiently than the direct fidelity estimation. Since the original protocol of anonymous quantum sensing cannot correctly estimate the amplitudes of the magnetic fields under state preparation errors, our results would improve the performance of anonymous quantum sensing in realistic situations.