General sandwiched Rényi marginal bound

Establish that, for every finite-dimensional bipartite state \(\rho_{AB}\), density operator \(\tau_A\), and order \(\alpha\in(1/2,1)\), the sandwiched Rényi divergence satisfies \(\widetilde D_{\alpha}(\rho_{AB}\|\tau_A\otimes\rho_B)\leq \frac{1}{\alpha}\min_{\sigma_B}\widetilde D_{\alpha}(\rho_{AB}\|\tau_A\otimes\sigma_B)\).

Background

The paper proves the corresponding marginal-versus-optimizer bound for the Petz–Rényi divergence for orders α∈[1/2,1)\alpha\in[1/2,1). For the sandwiched Rényi divergence, it establishes the inequality at α=1/2\alpha=1/2 for arbitrary states and for α∈(1/2,1)\alpha\in(1/2,1) when the bipartite state is pure or quantum-classical. The unresolved problem is to extend the same 1/α1/\alpha factor bound to arbitrary bipartite states in the range α∈(1/2,1)\alpha\in(1/2,1).

The authors note that a weaker factor 1/α21/\alpha^2 bound follows by combining the Petz–Rényi result with comparisons between the Petz and sandwiched Rényi divergences, but this does not resolve the conjectured sharper inequality.

References

Based on the results in #1{sec:fidelity}, we believe \begin{equation}\label{eq_conjec} \Tilde{D}{\alpha}(\rho{AB} | \tau_A \otimes \rho_B) \leq \frac{1}{\alpha} \min_{\sigma_B} \Tilde{D}{\alpha}(\rho{AB} | \tau_A \otimes \sigma_B) \end{equation} holds for general states as well. While being unable to show this in general, observe that a weaker bound follows from the Araki--Lieb--Thirring inequality and its reverse, which imply

eq_conjec:

$\Tilde{D}_{\alpha}(\rho_{AB} \| \tau_A \otimes \rho_B) \leq \frac{1}{\alpha} \min_{\sigma_B} \Tilde{D}_{\alpha}(\rho_{AB} \| \tau_A \otimes \sigma_B) $

— The marginal is pretty good  (2609.05225 - Schmitt et al., 4 Sep 2026) in Section Discussion, Eq. (\ref{eq_conjec})