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

Abstract

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 \(d\times d\) complex positive definite matrices. We prove that, for every fixed pair \(P,Q\in\Pd\), 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 \((-\infty,1]\) 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 \(z\ge1/2\) and of the Log-Euclidean fidelity as generalized fidelities. We also show that for \(d\ge2\) 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 \(P^{-1/2}Q^{1/2}W\) is similar to a positive definite matrix. This recovers the special-unitary constraint and disproves the global reverse inclusion \(SU(d)\) for \(d\ge2\).