A central task in quantum metrology is to exploit quantum correlations to outperform classical sensitivity limits. Metrologically useful entanglement is identified when the quantum Fisher information (QFI) exceeds a separability bound for a given parameter-encoding Hamiltonian. However, so far, only results for linear Hamiltonians are well established. Here, we characterize metrologically useful entanglement for nonlinear Hamiltonians, presenting separability bounds for collective angular momenta. Also, we provide a general expression for entangled states maximizing the QFI, which can be written as the superposition between the Greenberger-Horne-Zeilinger-like and singlet states. Finally, we compare the metrological usefulness of linear and nonlinear cases, in terms of entanglement detection and random symmetric states.

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• Entanglement detection
• Quantum Fisher information
• Quantum correlations in quantum information
• Quantum entanglement
• Quantum metrology