Studies of Symmetric Informationally-Complete Positive Operator Valued Measures for Quantum Information and Quantum Foundations

从来没有任何理论比量子力学更成功：在其一百多年的历史中，没有任何实验能与理论的预测相矛盾，直到最好的实验精度。量子力学的应用在现代世界随处可见，从DVD播放器中的微型激光器和数码相机中的传感器，到MRI扫描仪和核电站。具有讽刺意味的是，直到今天，理论核心的数学结构还没有得到很好的理解，在量子力学的物理解释中存在许多概念问题。我们想研究一个特殊的假设数学对象，称为量子态的对称信息完备集，简称SIC，它的对称性质可能为量子力学的数学结构提供重要的线索。SIC在量子贝叶斯主义计划中发挥着至关重要的作用，这是一种新开发的解释，旨在从概率论的完整机制中理解量子力学。在量子信息的各个领域中，也发现了SIC的有用性，例如，它们有助于提供更稳健的测量和更安全的通信。能够构建一般的SICs，甚至只是为了证明它们的存在，将给我们带来对量子力学的重要新理解以及量子信息的新应用，而且还没有人知道一旦完全理解量子力学，它还能给我们的生活带来什么。

There has never been any theory more successful than quantum mechanics: for over a hundred years in its history no experiment has ever contradicted the prediction of the theory up to the best experimental precision. Applications of quantum mechanics can be seen everywhere in the modern world, from tiny lasers in DVD players and sensors in digital cameras, to MRI scanners and nuclear power plants. Ironically, till today the mathematical structure lying at the heart of the theory is not well understood, and there are numerous conceptual issues in physical interpretations of quantum mechanics. We would like to propose the study of a special hypothetical mathematical object called a symmetric informationally-complete set of quantum states, or a SIC in short, whose symmetric properties could potentially provide important hints about the mathematical structure of quantum mechanics. SICs play a crucial role in the Quantum Bayesianism program, a newly developed interpretation aiming to understand quantum mechanics from the full machinery of probability theory. SICs have also been found useful in various areas in quantum information, for example they help provide more robust measurements and more secure communication. Being able to construct SICs in general, or even just to prove their existence, would bring us important new understandings about quantum mechanics as well as new applications in quantum information, and no one can yet know what more quantum mechanics can bring to our lives once it is fully understood.