The fidelity-based smooth min-relative entropy is a distinguishability measure that has appeared in a variety of contexts in prior work on quantum information, including resource theories like thermodynamics and coherence. Here we provide a comprehensive study of this quantity. First we prove that it satisfies several basic properties, including the data-processing inequality. We also establish connections between the fidelity-based smooth min-relative entropy and other widely used information-theoretic quantities, including smooth min-relative entropy and smooth sandwiched R\'enyi relative entropy, of which the sandwiched R\'enyi relative entropy and smooth max-relative entropy are special cases. After that, we use these connections to establish the second-order asymptotics of the fidelity-based smooth min-relative entropy and all smooth sandwiched R\'enyi relative entropies, finding that the first-order term is the quantum relative entropy and the second-order term involves the quantum relative entropy variance. Utilizing the properties derived, we also show how the fidelity-based smooth min-relative entropy provides one-shot bounds for operational tasks in general resource theories in which the target state is mixed, with a particular example being randomness distillation. The above observations then lead to second-order expansions of the upper bounds on distillable randomness, as well as the precise second-order asymptotics of the distillable randomness of particular classical-quantum states. Finally, we establish semi-definite programs for smooth max-relative entropy and smooth conditional min-entropy, as well as a bilinear program for the fidelity-based smooth min-relative entropy, which we subsequently use to explore the tightness of a bound relating the last to the first.

翻译：基于保真度的平滑最小相对熵是一种可区分性度量，已在量子信息学先前工作的多种场景中出现，包括热力学和相干性等资源理论。本文对此量进行了全面研究。首先，我们证明它满足若干基本性质，包括数据处理不等式。我们还建立了基于保真度的平滑最小相对熵与其他广泛使用的信息论量之间的联系，包括平滑最小相对熵和平滑夹层Rényi相对熵（其中夹层Rényi相对熵和平滑最大相对熵是其特例）。随后，我们利用这些联系推导了基于保真度的平滑最小相对熵及所有平滑夹层Rényi相对熵的二阶渐近特性，发现其一阶项为量子相对熵，二阶项涉及量子相对熵方差。基于所推导的性质，我们还展示了基于保真度的平滑最小相对熵如何为目标态混合的通用资源理论中的操作任务提供单发界限，随机性蒸馏即为一个具体实例。上述观察进而导致可蒸馏随机性上界的二阶展开式，以及特定经典-量子态可蒸馏随机性的精确二阶渐近特性。最后，我们建立了平滑最大相对熵和平滑条件最小熵的半定规划，以及基于保真度的平滑最小相对熵的双线性规划，并利用后者探讨了与前者相关的界限的紧致性。