Characterising the capacity region for a network can be extremely difficult. Even with independent sources, determining the capacity region can be as hard as the open problem of characterising all information inequalities. The majority of computable outer bounds in the literature are relaxations of the Linear Programming bound which involves entropy functions of random variables related to the sources and link messages. When sources are not independent, the problem is even more complicated. Extension of Linear Programming bounds to networks with correlated sources is largely open. Source dependence is usually specified via a joint probability distribution, and one of the main challenges in extending linear program bounds is the difficulty (or impossibility) of characterising arbitrary dependencies via entropy functions. This paper tackles the problem by answering the question of how well entropy functions can characterise correlation among sources. We show that by using carefully chosen auxiliary random variables, the characterisation can be fairly "accurate" Using such auxiliary random variables we also give implicit and explicit outer bounds on the capacity of networks with correlated sources. The characterisation of correlation or joint distribution via Shannon entropy functions is also applicable to other information measures such as Renyi entropy and Tsallis entropy.
Network Coding Capacity Regions via Entropy Functions
在本文中，我们使用熵函数来描述一般网络编码问题的速率容量元组集，这些元组可以是零解码错误，也可以是零解码错误。我们表明，当源是同位的，由Yeung，A First Course in信息论，第15.5节（2002年）得到的外边界是紧的，零误差可实现和消失误差可实现的速率容量元组的集合是相同的。我们还讨论了一组零错误和消失错误可实现的速率容量元组的网络编码问题的线性编码约束，路由约束（其中一些或所有节点只能执行路由）和保密约束。最后，我们表明，即使是显然简单的网络，最佳代码的设计可能是困难的。特别是，我们证明，对于增量组播问题和单源安全网络编码问题，表征的可实现集是非常困难的，线性网络代码可能不是最佳的。
Tight relations and equivalences between smooth relative entropies
RIKEN Hakubi Research Team, RIKEN Pioneering Research Institute (PRI) and RIKEN Center for Quantum Computing (RQC)（理化学研究所白兔研究团队、理学前驱性研究机构和理中心量子计算）；Scuola Normale Superiore（比萨高等师范学院）；QuSoft；Korteweg–de Vries Institute for Mathematics, University of Amsterdam（阿姆斯特丹大学科特维格-德弗里斯数学研究所）；Institute for Theoretical Physics, University of Amsterdam（阿姆斯特丹大学理论物理研究所）；Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge（剑桥大学数学科学中心应用数学与理论物理系）
RIKEN Hakubi Research Team, RIKEN Pioneering Research Institute (PRI) and RIKEN Center for Quantum Computing (RQC)（RIKEN 博慧研究团队、RIKEN 先锋研究所（PRI）和 RIKEN 量子计算中心）；Department of Electrical and Computer Engineering, National University of Singapore（新加坡国立大学电气与计算机工程系）；Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）
