Metrics induced by certain Hilbert-Schmidt fidelities on positive semi-definite matrices

Motivated by measuring the degree of similarity of a pair of quantum states (density matrices), we consider the metric property of the modified Bures angles and modified Bures distances of symmetric functions which are extensions of some fidelity measures on the spaces P of nonzero positive semi-definite matrices. We use the positive semi-definiteness of the Gram-type matrices to characterize the metric property of the modified Bures angles. As a consequence, we can show that the modified Bures angles induced by the geometric mean, harmonic mean, minimum and maximum of two positive numbers are metrics on P. In addition, we can also show that the metric property of the modified Bures angles is stronger than that of the modified Bures distances.