Experimental estimation of the quantum Fisher information from randomized measurements

The quantum Fisher information (QFI) represents a fundamental concept in quantum physics. On the one hand, it quantifies the metrological potential of quantum states in quantum-parameter-estimation measurements. On the other hand, it is intrinsically related to the quantum geometry and multipartite entanglement of many-body systems. Here, we explore how the QFI can be estimated via randomized measurements, an approach which has the advantage of being applicable to both pure and mixed quantum states. In the latter case, our method gives access to the sub-quantum Fisher information, which sets a lower bound on the QFI. We experimentally validate this approach using two platforms: a nitrogen-vacancy center spin in diamond and a 4-qubit state provided by a superconducting quantum computer. We further perform a numerical study on a many-body spin system to illustrate the advantage of our randomized-measurement approach in estimating multipartite entanglement, as compared to quantum state tomography. Our results highlight the general applicability of our method to general quantum platforms, including solid-state spin systems, superconducting quantum computers and trapped ions, hence providing a versatile tool to explore the essential role of the QFI in quantum physics.