Quantum Fisher Information and the Curvature of Entanglement

We explore the relationship between quantum Fisher information (QFI) and the negative of the second derivative of concurrence with respect to the coupling between two qubits, referred to as the curvature of entanglement (CoE). We analyze in detail the pure-state lossless case for which general results can be inferred and we also consider a simple interaction Hamiltonian in the case of one form of loss applied to the qubits. For a two-qubit quantum probe used to estimate the coupling constant appearing in the interaction Hamiltonian we show, for certain initial conditions, that there are times such that 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 several families of initially separable and initially entangled states, simple product measurements suffice to saturate the quantum Cramér-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.