  Recent advancements in entanglement-based quantum metrology have been
significant. A fundamental connection between generalized cat states and
sensitivity in quantum metrology has recently been established. Generalized cat
states are characterized by an index indicating coherence among macroscopically
distinct states. This criterion enables the identification of diverse states as
generalized cat states, encompassing classical mixtures of exponentially large
numbers of states. However, preparing large generalized cat states remains
challenging with current technology. Here we propose a protocol to generate
metrologically useful cat states through repetitive measurements on a quantum
spin system of N spins, which we call a spin ensemble. The states used as
sensors to beat the classical limit are called the metrologically useful cat
states, which are well characterized by the index to indicate the coherence
between macroscopically distinct states. When the spin ensemble is collectively
coupled with an ancillary qubit, it allows for the read out of its total
magnetization. Starting from a thermal equilibrium state of the spin ensemble,
we demonstrate that we can increase the coherence between the spin ensemble via
repetitive measurements of the total magnetization using the ancillary qubit.
Notably, our method for creating the metrologically useful cat states requires
no control over the spin ensemble. As a potential experimental realization, we
discuss a hybrid system composed of a superconducting flux qubit and donor
spins in silicon. Our results pave the way for the realization of the
entanglement-enhanced quantum metrology.
