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Trace-Norm Overlaps of Quantum States: Interpolation, Equality, and Data-Processing Rigidity — arXiv:2609.14203

Source

arXiv (en)

Original

https://arxiv.org/abs/2609.14203

License (as archived)

CC0-1.0

Archived

2026-10-16T02:24:19.623Z

SHA-256 of the archived text

2d9bba561c3bdbb0ca3adb2fcb8959bc95d077d5d9bdc62c0d86c5fb082a8eae

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Trace-Norm Overlaps of Quantum States: Interpolation, Equality, and Data-Processing Rigidity — arXiv:2609.14203 — arXiv (en), https://arxiv.org/abs/2609.14203 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_65256da1854a (Metraipolis archive dc_65256da1854a, sha256 2d9bba561c3bdbb0ca3adb2fcb8959bc95d077d5d9bdc62c0d86c5fb082a8eae)

@misc{dc65256da1854a,
  title = {Trace-Norm Overlaps of Quantum States: Interpolation, Equality, and Data-Processing Rigidity — arXiv:2609.14203},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.14203},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_65256da1854a; sha256 2d9bba561c3bdbb0ca3adb2fcb8959bc95d077d5d9bdc62c0d86c5fb082a8eae},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.14203. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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.

Tema degli esploratori: fisica e astronomia (retro-catalogo arXiv: annunci del 2026-09-15 in quant-ph).
Categorie arXiv: Quantum Physics (quant-ph) ; Mathematical Physics (math-ph).
Pagina del lavoro: https://arxiv.org/abs/2609.14203 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-16T01:57:57.012Z dalla pagina https://arxiv.org/catchup/quant-ph/2026-09-15?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).