Smooth min-entropy lower bounds for approximation chains
share
For a state ρ_A_1^n B, we call a sequence of states (σ_A_1^k B^(k))_k=1^n an approximation chain if for every 1 ≤ k ≤ n, ρ_A_1^k B≈_ϵσ_A_1^k B^(k). In general, it is not possible to lower bound the smooth min-entropy of such a ρ_A_1^n B, in terms of the entropies of σ_A_1^k B^(k) without incurring very large penalty factors. In this paper, we study such approximation chains under additional assumptions. We begin by proving a simple entropic triangle inequality, which allows us to bound the smooth min-entropy of a state in terms of the Rényi entropy of an arbitrary auxiliary state while taking into account the smooth max-relative entropy between the two. Using this triangle inequality, we create lower bounds for the smooth min-entropy of a state in terms of the entropies of its approximation chain in various scenarios. In particular, utilising this approach, we prove an approximate version of entropy accumulation and also provide a solution to the source correlation problem in quantum key distribution.
READ FULL TEXT