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：在它的一百多年历史中，没有任何实验能够达到最好的实验精度，与理论的预言相矛盾。量子力学的应用在现代世界随处可见，从DVD播放机中的微型激光器和数码相机中的传感器，到核磁共振扫描仪和核电站。具有讽刺意味的是，直到今天，这个理论的核心数学结构还没有被很好地理解，在量子力学的物理解释中还有许多概念问题。我们建议研究一种特殊的假设数学对象，称为对称信息完备量子态集，或简称SIC，其对称特性可能为量子力学的数学结构提供重要线索。量子贝叶斯理论是一种新发展的解释，旨在从概率论的完整机制中理解量子力学。sic也被发现在量子信息的各个领域都很有用，例如，它们有助于提供更可靠的测量和更安全的通信。能够构建一般的量子力学，甚至只是证明它们的存在，将给我们带来对量子力学的重要新理解，以及在量子信息中的新应用，而且没有人知道一旦量子力学被完全理解，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.