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 approximate versions of the asymptotic equipartition property and entropy accumulation. In our companion paper, we show that the techniques developed in this paper can be used to prove the security of quantum key distribution in the presence of source correlations.

翻译：对于状态 $\rho_{A_1^n B}$，若存在序列 $(\sigma_{A_1^k B}^{(k)})_{k=1}^n$ 满足对每个 $1 \leq k \leq n$ 均有 $\rho_{A_1^k B} \approx_\epsilon \sigma_{A_1^k B}^{(k)}$，则称该序列为近似链。一般而言，若不引入较大的惩罚因子，无法仅通过 $\sigma_{A_1^k B}^{(k)}$ 的熵来下界界定 $\rho_{A_1^n B}$ 的平滑最小熵。本文在附加假设下研究此类近似链。首先证明一个简单的熵三角不等式，该不等式允许我们在考虑状态间平滑最大相对熵的前提下，通过任意辅助状态的Rényi熵来界定该状态的平滑最小熵。基于此三角不等式，我们针对不同场景，建立了通过近似链熵值来界定状态平滑最小熵的下界。特别地，利用该方法，我们证明了渐近等分性质和熵累积的近似版本。在配套论文中，我们将展示本文发展的技术可用于证明存在源关联时量子密钥分配的安全性。