Quantum Fisher information density in infinite-size extended Ising chains

In this paper, we investigate quantum Fisher information (QFI) density in the ground state of the one-dimensional infinite-size extended quantum Ising model, a system known for its rich phase diagram and topological quantum phase transitions. Notably, the QFI density itself displays clear signatures at many critical points, making it a better indicator compared to previously used two-qubit QFI (which depends upon a two-qubit reduced density matrix). This advantage stems from the QFI density’s reliance on all two-qubit reduced density matrices in the ground state, rather than just one. Beyond critical phenomena, we explore the connection between QFI density and quantum entanglement. We identify wide regions where metrologically useful entanglement is present and consequently quantum-enhanced metrology is expected. Furthermore, the QFI density shows peaks in the vicinity of some critical points, suggesting the possibility of criticality-enhanced metrology. Overall, our results demonstrate that the QFI density serves as a powerful tool for characterizing both quantum criticality and metrologically useful entanglement in the extended quantum Ising model, offering valuable insights for both theoretical understanding and future experimental investigations in quantum metrology and quantum criticality.