For a state $\rho_{A_1^n B}$, we call a sequence of states $(\sigma_{A_1^k B}^{(k)})_{k=1}^n$ an approximation chain if for every $1 \leq k \leq n$, $\rho_{A_1^k B} \approx_\epsilon \sigma_{A_1^k B}^{(k)}$. In general, it is not possible to lower bound the smooth min-entropy of such a $\rho_{A_1^n B}$, in terms of the entropies of $\sigma_{A_1^k B}^{(k)}$ 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 R\'enyi 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.

翻译：对于态 $\rho_{A_1^n B}$，如果对每个 $1 \leq k \leq n$ 都有 $\rho_{A_1^k B} \approx_\epsilon \sigma_{A_1^k B}^{(k)}$，则称态序列 $(\sigma_{A_1^k B}^{(k)})_{k=1}^n$ 为一条近似链。一般情况下，若不引入极大的惩罚因子，则无法利用 $\sigma_{A_1^k B}^{(k)}$ 的熵来给出 $\rho_{A_1^n B}$ 的光滑最小熵下界。本文在额外假设下研究此类近似链。我们首先证明一个简单的熵三角不等式，该不等式可在考虑两个态之间的光滑最大相对熵的条件下，通过一个任意辅助态的Rényi熵来界定该态的光滑最小熵。利用这一三角不等式，我们针对不同场景，基于近似链中各态的熵建立了态的光滑最小熵下界。特别地，通过这一方法，我们证明了熵积累的近似版本，并解决了量子密钥分发中的源相关问题。