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

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-