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 $\text{Pd}$, the cone of $d\times d$ complex positive definite matrices. We prove that, for every fixed pair $P,Q\in \text{Pd}$, the one-sided polar fidelities $x\mapsto F_{P^x}(P,Q)$ and $x\mapsto F_{Q^x}(P,Q)$, as well as their symmetrization, are nondecreasing on $(-\mathrm{\infty },1]$ and nonincreasing on $[1,\mathrm{\infty })$. Hence the polar paths realize exactly the interval $[\text{FM}(P,Q),\text{FU}(P,Q)]$ on $[-1,1]$, yielding pointwise, generally pair-dependent realizations of all $z$-fidelity values with $z\ge 1\mathrm{/}2$ and of the Log-Euclidean fidelity as generalized fidelities. We also show that for $d\ge 2$ and $0<z<1\mathrm{/}2$, such realization fails in general, even for interior fidelities. We further solve the fixed-pair Holevo-base equation $F_R(P,Q)=\text{FH}(P,Q)$, 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 $P^{-1\mathrm{/}2}{Q}^{1\mathrm{/}2}W$ is similar to a positive definite matrix. This recovers the special-unitary constraint and disproves the global reverse inclusion $SU(d)$ for $d\ge 2$.