Polar Fidelities, Holevo Bases, and Unitary Factors of Generalized Fidelity

Motivated by several open problems in the Afham-Ferrie theory of generalized fidelity, we study polar fidelities, Holevo bases, and unitary factors on Pd, the cone of dtimes d complex positive definite matrices. We prove that, for every fixed pair P,QinPd, the one-sided polar fidelities xmapsto F_P^x(P,Q) and xmapsto F_Q^x(P,Q), as well as their symmetrization, are nondecreasing on (-infty,1] and nonincreasing on \[1,infty). Hence the polar paths realize exactly the interval [FM(P,Q),FU(P,Q)\] on [-1,1], yielding pointwise, generally pair-dependent realizations of all z-fidelity values with zge1/2 and of the Log-Euclidean fidelity as generalized fidelities. We also show that for dge2 and 0<z<1/2, such realization fails in general, even for interior fidelities. We further solve the fixed-pair Holevo-base equation F_R(P,Q)=FH(P,Q), classify all Holevo bases, and classify exactly which unitary factors of generalized fidelity can arise from a base R. These are precisely the unitaries W for which P^-1/2Q^1/2W is similar to a positive definite matrix. This recovers the special-unitary constraint and disproves the global reverse inclusion SU(d) for dge2.