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

Abstract

Let $ρ$ and $σ$ be density operators on a separable Hilbert space. For $0<α<1$, we study the directed trace-norm overlap $Φ_α(ρ,σ)=\|ρ^ασ^{1-α}\|_1$ and its symmetrized form $\mathcal F_α(ρ,σ)=\frac12\bigl(Φ_α(ρ,σ)+Φ_{1-α}(ρ,σ)\bigr)$. At $α=\tfrac12$, 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 $α\mapstoΦ_α(ρ,σ)$ is log-convex. This yields a sharp lower bound for $\mathcal F_α$ in terms of the root fidelity, together with a stronger intermediate geometric-mean bound. We also show that $Φ_α$ is the trace functional $Q_{α,1/2}$ associated with the $α$-$z$ Renyi divergence, which connects the directed overlap with the known $α$-$z$ Renyi theory. We then determine the exact data-processing behavior of the symmetrized family. Universal monotonicity under quantum channels holds only at $α=\tfrac12$; for every other value of $α$, 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 $ρ$ and $σ$ commute, without requiring either faithfulness or any additional support assumption.