Quantum Fisher Information and Curvature of Entanglement in Higher Dimensions

We explore the relationship between quantum Fisher information (QFI) and the second derivative of entanglement with respect to the coupling between qubits, referred to as the curvature

of entanglement (CoE). For a two-qubit quantum probe used to estimate the coupling constant appearing in a simple interaction Hamiltonian, choosing concurrence as the entanglement measure we show that at certain times CoE = −QFI. These times can be associated with the concurrence, viewed as a function of the coupling parameter, being

a maximum. We examine the time evolution of the concurrence of the eigenstates of the symmetric logarithmic derivative and show that, for both initially separable and initially entangled states, simple product measurements suffice to saturate the quantum Cramer-Rao bound when CoE = −QFI, while otherwise, in general, entangled measurements are required giving an operational significance to the points in time when CoE = −QFI. We investigate similar quantitites for three qubit system with three-tangle being used as the entanglement measure. We find similar relationships in this higher system with CoE= - γ Qfi with γ  being a scaling factor determined by the choice of interaction Hamiltonian.