A Weighted Spectral Quantum Fidelity

We introduce and study a one-parameter family of fidelity-type quantities based on the weighted spectral geometric mean, which we call the \emph{weighted spectral fidelity} $\mathsf{F}_t^{\mathrm{spec}}(\rho,\sigma):=\Tr\!\big[\rho(\rho^{-1}\sharp\sigma)^{2t}\big],\ t\in[0,1].$ This family interpolates smoothly between the trivial overlap ($t=0,1$) and the Uhlmann (root) fidelity at $t=\tfrac12$, and it is distinct from the sandwiched R\'enyi family except at this midpoint. We establish core structural features-unitary invariance, tensor stabilization and multiplicativity, flip symmetry, endpoint behavior, and a orthogonality criterion. We further show explicit \emph{violations of DPI} for generic $t\neq\tfrac12$. For concavity in the state variables we obtain concavity in each variable separately. Closed forms are obtained for pure states and for qubits in Bloch coordinates. We also extend the first Fuchs--van de Graaf inequality to $\mathsf{F}_t^{\mathrm{spec}}$ for all $t\in[0,1]$, while the second inequality fails away from the midpoint.