Entropy inequalities and entropic uncertainty relations

熵在从凝聚态物理到宇宙学的许多物理学领域中起着基础性的作用。量子信息理论的最新发展通过提供对例如冯·诺依曼熵作为数据压缩率的操作性解释，改善了对熵的理解。系统各部分熵之间的不等式是熵函数最基本的性质之一，一个著名的例子是冯诺依曼熵的强次可加性。与互补观测量的测量熵相关的不等式，即所谓的熵不确定性关系，首先在物理学基础的背景下被考虑，已经成功地用于量子密码的安全证明。这个项目将研究冯·诺依曼熵的不等式，更一般的是量子雷尼熵。本研究有两个主要目标：第一，决定是否存在进一步的冯诺依曼熵不等式（所有已知的不等式减少到强次可加性）。第二，将熵的不确定关系推广到条件量子Rényi熵的不确定关系。该研究对自旋系统的熵极限和纠缠熵的研究具有重要意义，并将直接影响量子信息论、密码学和量子力学基础的研究。

Entropy plays a fundamental role in many areas of physics ranging from condensed matter physics to cosmology. Recent developments in quantum information theory have improved the understanding of entropy by providing operational interpretations of, for instance, von Neumann entropy as data compression rate. Inequalities that relate the entropies of parts of a system are the among the most elementary properties of entropy functions, a well-known example is the strong subadditivity of von Neumann entropy. Inequalities that relate the entropies of measurements of complementary observables, so-called entropic uncertainty relations, first considered in the context of the foundations of physics, have been successfully employed in security proofs of quantum cryptography. This project will study inequalities for the von Neumann entropy and more generally for quantum Rényi entropies. This study has two main goals: First, to decide whether or not there exist further inequalities of the von Neumann entropy (all known inequalities reduce to strong subadditivity). Second, to generalise the entropic uncertainty relations to uncertainty relations for conditional quantum Rényi entropies. The proposed research is of significance to the study of entropy limits and entanglement entropies in spin systems, and will directly impact the study of quantum information theory, cryptography and the foundations of quantum mechanics.