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 divergence of order $α\in[1/2,1)$, replacing the optimizing state on $B$ by the marginal $ρ_B$ results in a multiplicative overhead of at most $1/α$. We also show a similar relation for the fidelity, and in the case of pure or quantum-classical states for the sandwiched Rényi divergence.