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 ${\Phi }_{\alpha }(\rho ,\sigma )=\Vert {\rho }^{\alpha }{\sigma }^{1-\alpha }{\Vert }_1$ and its symmetrized form ${\mathcal{F}}_{\alpha }(\rho ,\sigma )=\frac{1}{2}({\Phi }_{\alpha }(\rho ,\sigma )+{\Phi }_{1-\alpha }(\rho ,\sigma ))$. At $\alpha =\frac{1}{2}$, 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 $\alpha =\frac{1}{2}$; 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.

26 pages