Trace-Norm Overlaps of Quantum States: Interpolation, Equality, and Data-Processing Rigidity

Abstract

Let $\rho$ and $\sigma$ be density operators on a separable Hilbert space. For $0<\alpha<1$, we study the directed trace-norm overlap $\Phi_\alpha(\rho,\sigma)=\|\rho^\alpha\sigma^{1-\alpha}\|_1$ and its symmetrized form $\mathcal F_\alpha(\rho,\sigma)=\frac12\bigl(\Phi_\alpha(\rho,\sigma)+\Phi_{1-\alpha}(\rho,\sigma)\bigr)$. At $\alpha=\tfrac12$, both quantities coincide with the root Uhlmann--Jozsa fidelity. Our aim is not to introduce a new notion of fidelity, but to understand how these overlaps vary with the parameter and when equality occurs in the resulting inequalities. We first prove that $\alpha\mapsto\Phi_\alpha(\rho,\sigma)$ is log-convex. This yields a sharp lower bound for $\mathcal F_\alpha$ in terms of the root fidelity, together with a stronger intermediate geometric-mean bound. We also show that $\Phi_\alpha$ is the trace functional $Q_{\alpha,1/2}$ associated with the $\alpha$-$z$ Renyi divergence, which connects the directed overlap with the known $\alpha$-$z$ Renyi theory. We then determine the exact data-processing behavior of the symmetrized family. Universal monotonicity under quantum channels holds only at $\alpha=\tfrac12$; for every other value of $\alpha$, it already fails under diagonal pinching of faithful real qubit states. In finite dimensions, we give a complete spectral description of the equality cases and provide explicit noncommuting examples. We also discuss some natural questions about overlap-preserving maps. Finally, we prove that the difference between the Petz overlap and the directed trace-norm overlap vanishes exactly when $\rho$ and $\sigma$ commute, without requiring either faithfulness or any additional support assumption.