Experimental benchmark of pseudo-Hermitian qubit sensing via Naimark dilation

Abstract

Non-Hermitian dynamics can sharply amplify the response of a quantum sensor near eigenstate coalescence, but whether this amplification yields a genuine metrological gain depends on how the non-unitary evolution is physically realised and how its resources are counted. We address this question experimentally by implementing the pseudo-Hermitian qubit sensing protocol of Chu et al . [Phys. Rev. Lett. 124, 020501 (2020)] on a two-qubit nuclear magnetic resonance processor via Naimark dilation. Because both the postselected signal and the full two-qubit probability are obtained from a single set of population measurements, the enhanced response and its postselection cost are accessed within the same experiment. We observe the characteristic steep response of the postselected branch, whereas the sensitivity extracted from the absolute two-qubit probability does not surpass the Ramsey benchmark for the same sensor encoding, consistent with the theoretical resource bound derived by Ding et al . [Phys. Rev. Lett. 131, 160801 (2023)] for the corresponding dilation scheme. The experiment complements the recent photonic implementation of the direct non-unitary route and provides an experimental benchmark of the Hermitian-dilation route with explicit resource accounting.