我们给出了相对熵的一个新的特征，也被称为Kullback-Leibler发散。我们使用了一些与概率论有关的有趣的分类。特别是，我们考虑一个类别FinStat的对象是一个有限集配备了概率分布，而态射是一个测度保持功能$f: X \to Y$连同随机右逆$s: Y \to X$。函数$f$可以被认为是一个测量过程，而s提供了一个假设，关于给定测量结果的被测系统的状态。给定该数据，我们可以定义$X$上的概率分布相对于通过沿着$s$向前推动$Y$上的概率分布而给出的“先验”的熵。如果这些分布一致，我们说$s$是“最优”的。我们发现，任何凸线性，较低的连续函子从FinStat的添加剂幺半群$[0,\infty]$消失时，$s$是最佳的必须是这个相对熵的标量倍。我们的证明是独立的所有早期的特征，但灵感来自工作的Petz。
英文摘要：
We give a new characterization of relative entropy, also known as the Kullback-Leibler divergence. We use a number of interesting categories related to probability theory. In particular, we consider a category FinStat where an object is a finite set equipped with a probability distribution, while a morphism is a measure-preserving function $f: X \to Y$ together with a stochastic right inverse $s: Y \to X$. The function $f$ can be thought of as a measurement process, while s provides a hypothesis about the state of the measured system given the result of a measurement. Given this data we can define the entropy of the probability distribution on $X$ relative to the "prior" given by pushing the probability distribution on $Y$ forwards along $s$. We say that $s$ is "optimal" if these distributions agree. We show that any convex linear, lower semicontinuous functor from FinStat to the additive monoid $[0,\infty]$ which vanishes when $s$ is optimal must be a scalar multiple of this relative entropy. Our proof is independent of all earlier characterizations, but inspired by the work of Petz.
Haar-Bayesian Pure-State Prediction under Relative-Entropy Loss: Arbitrary-Effect Reduction and Global Optimality
School of Data Science, The Chinese University of Hong Kong（香港中文大学数据科学学院）；International Quantum Academy (IQA)（国际量子学院）；Indian Statistical Institute（印度统计研究所）
Semidefinite optimization of the quantum relative entropy of channels
Institute for Quantum Information, RWTH Aachen University（亚琛工业大学量子信息研究所）；School of Electrical and Computer Engineering, Cornell University（康奈尔大学电气与计算机工程学院）
