Estimating entanglement in a class of N-qudit states

The logarithmic derivative (or, quantum score) of a positive definite density matrix appearing in the quantum Fisher information is discussed, and its exact expression is presented. Then, the problem of estimating the parameters in a class of the Werner-type N-qudit states is studied in the context of the quantum Cram'er-Rao inequality. The largest value of the lower bound to the error of estimate by the quantum Fisher information is shown to coincide with the separability point only in the case of two qubits. It is found, on the other hand, that such largest values give rise to the universal fidelity that is independent of the system size.