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.

arXiv:2609.14203 [quant-ph]

(or arXiv:2609.14203v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.14203 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.