纠缠相对熵的可达性与下半连续性，以及相关主题的变体
Attainability and lower semi-continuity of the relative entropy of entanglement, and variations on the theme
AI总结：
本文证明了纠缠相对熵在无限维下也具有可达性与下半连续性，并推导出其对偶变分表达式，该结果适用于多种资源量化指标，关键在于自由状态生成锥的弱*闭性，同时给出了能量约束下的连续性估计。
摘要：
纠缠相对熵 $E_R$ 定义为多体量子态与可分态集合之间的距离，由量子相对熵来度量。我们证明该优化总是可以实现的，即任何状态都存在一个最近可分态，即使在无限维情况下也是如此；此外，$E_R$ 处处具有下半连续性。我们利用这一点推导出了 $E_R$ 的对偶变分表达式，用外部上确界代替下确界。这些迄今为止似乎未被注意到的结果，不仅适用于纠缠相对熵及其多体推广，还适用于许多其他类似的资源量化指标，例如非高斯性、非经典性、Wigner 负性的相对熵——更一般地，适用于所有与非负 $λ$ 准概率分布状态集合之间的相对熵距离。支撑所有这些应用的关键假设是由自由状态生成的锥的弱*闭性，因此我们开发的技术涉及一系列经典的泛函分析结果。我们通过在存在能量约束的情况下，为 $E_R$ 及其密切相关的量提供显式且渐近紧致的连续性估计，补充了我们的分析。
英文摘要：
The relative entropy of entanglement $E_R$ is defined as the distance of a multi-partite quantum state from the set of separable states as measured by the quantum relative entropy. We show that this optimisation is always achieved, i.e. any state admits a closest separable state, even in infinite dimensions; also, $E_R$ is everywhere lower semi-continuous. We use this to derive a dual variational expression for $E_R$ in terms of an external supremum instead of infimum. These results, which seem to have gone unnoticed so far, hold not only for the relative entropy of entanglement and its multi-partite generalisations, but also for many other similar resource quantifiers, such as the relative entropy of non-Gaussianity, of non-classicality, of Wigner negativity $\unicode{8212}$ more generally, all relative entropy distances from the sets of states with non-negative $λ$-quasi-probability distribution. The crucial hypothesis underpinning all these applications is the weak*-closedness of the cone generated by free states, and for this reason the techniques we develop involve a bouquet of classical results from functional analysis. We complement our analysis by giving explicit and asymptotically tight continuity estimates for $E_R$ and closely related quantities in the presence of an energy constraint.
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