Detecting large quantum Fisher information with finite measurement precision

We propose an experimentally accessible scheme to determine lower bounds on the quantum Fisher information (QFI), which ascertains multipartite entanglement or usefulness for quantum metrology. The scheme is based on comparing the measurement statistics of a state before and after a small unitary rotation. We argue that, in general, limited resolution of collective observables prevents the detection of large QFI. This can be overcome by performing an additional operation prior to the measurement. We illustrate the power of this protocol for present-day spin-squeezing experiments, where the same operation used for the preparation of the initial spin-squeezed state improves the lower bound on the QFI by two orders of magnitude. We also establish a connection to Leggett-Garg inequalities. We show how to simulate a variant of the inequalities with our protocol and demonstrate that a large QFI is necessary for their violation with coarse-grained detectors.