联合量子测量、非经典关联及量子信息相关研究

量子测量的联合可测性问题是量子力学基础研究的重要问题之一，它与称为量子导引的一种重要的非经典关联有密切联系。在一定条件下，量子导引与一簇量子测量能否联合可测之间有一对一的对应关系，即一簇量子测量不能联合测量当且仅当它们可以用于展示量子导引。本项目通过将联合可测性问题转化为凸集问题，从而用凸分析和几何泛函分析为工具，来深入研究如下联合可测性和量子导引等相关问题：(1)量子测量簇联合可测的凸几何刻画、量子导引的判据以及二者之间的相互转化、联合测量与张量分解的关系；(2)局部隐变量/隐藏态模型在量子理论中的局限性的大小；(3)随机量子系统导引及其他非经典关联的某些统计性质；(4)基于高斯随机矩阵指数所生成的连续随机量子测量之间的联合可测性的大小估计。

The joint measurability problem of quantum measurements is one of important problems in fundamental study of Quantum Mechanics. It is intimately related to one kind of special quantum correlations which is called quantum steering. There is one-to-one mapping between steering and joint measurability problems under some conditions, i.e., a collection of quantum measurements cannot be jointly measured if and only if they can be used for demonstrating quantum steering. In this program, we will study the open problem of joint measurability of quantum measurements, that is, characterize the geometric version of the joint measurability of a collection of quantum measurements by using tools from convex geometry and geometric functional analysis. (1)We will characterize the joint measurability from convex geometry, present criteria for quantum steering via the inter-convention between the joint measurability problem and quantum non-locality, we also investigate the relationship between the joint measurability of quantum measurements and the decomposition of a doubly normalized tensor; and (2)we make a comparative study of volumes of the sets of classical correlation matrices and quantum correlation matrices; (3)we also consider the steering and the statistical properties of non-classical correlation of a random state; finally (4)we estimate the probability of joint measurability of two independent random quantum measurements, generated by the matrix exponential of random Gaussian matrix.