For zero-mean Gaussian states, path entanglement reduces the quantum Fisher information for phase shifts, and the optimal states are unentangled, properly ordered squeezed states.
704-706 in [61] to an infinite-dimensional set- ting by noting that density operators are trace class and therefore compact ([62] Theorem VI.21)
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A Trade-Off Between Path Entanglement and Quantum Sensitivity
For zero-mean Gaussian states, path entanglement reduces the quantum Fisher information for phase shifts, and the optimal states are unentangled, properly ordered squeezed states.