量子信息科学中的凸优化问题研究

The rapid development of quantum information science is plagued with complex mathematical problems, and convex optimization theory is the appropriate tool to tackle them. In quantum mechanics, since the set of quantum states as well as quantum measurements are defined by semi-definite constraints, many important problems in quantum information like the computation of ground state energies, local value of the Bell inequality, entanglement detection and quantification, and so on can all be solved using convex optimization techniques. Semi-definite programming or even linear programming methods are proved to be very useful for small-scale problems. However, along with the rapid development of quantum information science, the problems we need to solve become more and more complex, thus new and more efficient convex optimization techniques are urgently required. In this project, by making use of the so-called Gilbert’s algorithm which has been successfully applied to quantum information very recently, we will first study how to perform convex optimization over the space of quantum measurements, and then extend it to the joint space of quantum states and quantum measurements. Including our previous results on the convex optimization over the space of quantum states, we will eventually develop a complete and systematic set of convex optimization techniques, hopefully to cover most of the convex optimization tasks in quantum information science.

复杂的数学问题一直困扰着量子信息科学的快速发展，而凸优化理论是解决这些问题最恰当的工具。在量子力学中，由于量子态的集合以及量子测量的集合都是由半正定的约束条件来定义的，因此量子信息中的许多重要问题比如基态能量的计算、贝尔不等式的局域值以及纠缠的判定和度量等都可以用凸优化的技术来解决。半定规划甚至线性规划的方法对于小规模的问题非常有效，然而随着量子信息科学的快速发展，我们所需解决的问题越来越复杂，因此新的更高效的凸优化技术亟待发展。在本项目中，我们依托最近被成功应用到量子信息领域的吉尔伯特算法，首先研究如何在量子测量的空间上进行凸优化，进而将其拓展到量子态和量子测量的联合空间上。包括我们前期在量子态空间凸优化问题已经取得的成果，最终我们将发展一套完整且系统的凸优化技术，基本能够涵盖量子信息领域大部分的凸优化任务。

随着量子信息科学的快速发展，我们所需解决的问题越来越复杂，因此新的更高效的凸优化算法亟待发展。在本项目中，我们依托最近被成功应用到量子信息领域的吉尔伯特算法，首先将吉尔伯特算法拓展到了量子测量空间，并将吉尔伯特算法用作满足限制条件的工具，完成了在量子测量空间以及在量子态和量子测量联合空间上的凸优化任务。包括我们前期在量子态空间凸优化问题已经取得的成果，最终我们发展了一套完整且系统的凸优化技术，基本能够涵盖量子信息领域大部分的凸优化任务。

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