The marginal is pretty good

Abstract

One-shot information theory measures often require an optimization over states, but the form of these optimizers can be complicated or depend on the initial problem in nonlinear ways. In this note, we show that in many instances using the marginal instead of the optimal state is sufficiently good and only changes the result by a small factor. We prove that for the Petz--Rényi and sandwiched--Rényi divergences of order $\alpha \in (0,1)$, replacing the optimizing state on $B$ by the marginal ${\rho }_B$ results in a multiplicative overhead of at most $1/\alpha$.

12 pages, Gui-Long Jiang and Shi-Bing Li added as authors, Theorem 3.1 now holds for $\alpha \in (0,1)$ (instead of $[1/2,1)$), Theorem 4.1 and 4.2 show the same statement for the sandwiched Rényi divergence. This generalizes the previous result that only worked for $\alpha =1/2$ and special classes of states