论文标题
爱泼斯坦凹度定理和相关不平等的单调性版本
Monotonicity versions of Epstein's Concavity Theorem and related inequalities
论文作者
论文摘要
许多迹线不等式可以表示为凹度/凸定理，也可以作为单调定理表示。一个经典的例子是量子相对熵的关节凸度，这等同于数据处理不等式。后者说，量子操作永远不会增加相对熵。如刚才提到的示例，单调性版本通常具有许多优势，并且通常具有直接的物理应用。此外，单调性结果通常对比量子操作（完全为正的）更大的一类地图有效。在本文中，我们证明了几个新的单调性结果，第一个结果是单调性定理，它是一个简单的推论，是爱泼斯坦著名的凹入定理。我们的起点是Lieb凹陷和Lieb凸定理的单调性版本。我们还使用插值以它们的一般形式提供了两个新的证据。然后，我们通过几个双重性论点证明了我们的新单调定理。
Many trace inequalities can be expressed either as concavity/convexity theorems or as monotonicity theorems. A classic example is the joint convexity of the quantum relative entropy which is equivalent to the Data Processing Inequality. The latter says that quantum operations can never increase the relative entropy. The monotonicity versions often have many advantages, and often have direct physical application, as in the example just mentioned. Moreover, the monotonicity results are often valid for a larger class of maps than, say, quantum operations (which are completely positive). In this paper we prove several new monotonicity results, the first of which is a monotonicity theorem that has as a simple corollary a celebrated concavity theorem of Epstein. Our starting points are the monotonicity versions of the Lieb Concavity and the Lieb Convexity Theorems. We also give two new proofs of these in their general forms using interpolation. We then prove our new monotonicity theorems by several duality arguments.
