We study the relative entropy in QFT comparing the vacuum state to a special family of purifications determined by an input state and constructed using relative modular flow. We use this to prove a conjecture by Wall that relates the shape derivative of relative entropy to a variational expression over the averaged null energy (ANE) of possible purifications. This variational expression can be used to easily prove the quantum null energy condition (QNEC). We formulate Wall’s conjecture as a theorem pertaining to operator algebras satisfying the properties of a half-sided modular inclusion, with the additional assumption that the input state has finite averaged null energy. We also give a new derivation of the strong superadditivity property of relative entropy in this context. We speculate about possible connections to the recent methods used to strengthen monotonicity of relative entropy with recovery maps.

中文翻译：

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从 ANEC 恢复 QNEC

我们研究了 QFT 中的相对熵，将真空状态与由输入状态确定并使用相对模块化流构建的特殊净化系列进行比较。我们用它来证明 Wall 的一个猜想，该猜想将相对熵的形状导数与可能纯化的平均零能量 (ANE) 的变分表达式相关联。此变分表达式可用于轻松证明量子零能量条件 (QNEC)。我们将 Wall 猜想表述为与满足半边模包含性质的算子代数有关的定理，并附加假设输入状态具有有限的平均零能量。在这种情况下，我们还给出了相对熵的强超可加性属性的新推导。