The mathematical theory of optimization has become a vital tool in various branches of science. Especially, the so-called semidefinite programs (SDPs) as a special instance of convex optimization problems have found widespread applications in quantum information. In many applications, however, one cannot directly formulate an SDP, as some intractable rank constraints remain on the occurring matrices. In this talk, we establish a link between rank-constrained optimization and the theory of quantum entanglement. More precisely, we prove that a large class of rank-constrained semidefinite programs can be written as a convex optimization over separable quantum states and, consequently, we construct a complete hierarchy of semidefinite programs for solving the original problem. This hierarchy not only provides a sequence of certified bounds for the rank-constrained optimization problem, but also gives pretty good and often exact values in practice when the lowest level of the hierarchy is considered. We demonstrate that our approach can be used for relevant problems in quantum information processing, such as the quantum marginal problem, the existence of absolutely maximally entangled states, the optimization over pure states, the characterization of faithful entanglement, and the dimension witness of quantum systems.

References

[1] X.-D. Yu, T. Simnacher, N. Wyderka, H. C. Nguyen, and O. Gühne, Nat. Commun. 12, 1012 (2021).

[2] X.-D. Yu, T. Simnacher, H. C. Nguyen, and O. Gühne, PRX Quantum 3, 010340 (2022).

[3] O. Gühne, Y. Mao, and X.-D. Yu, Phys. Rev. Lett. 126, 140503 (2021).

[4] X.-D. Yu, I. Veeren, and O. Gühne, Phys. Rev. A 109, L030201 (2024).

报告人简介

于晓东，山东大学物理学院教授，博士生导师。2017年在山东大学获得理论物理博士学位，2017-2021年在德国锡根大学进行博士后研究，2021年10月受聘回到山东大学工作，先后入选山东大学齐鲁青年学者（第一层次）、山东省泰山学者青年专家、国家海外高层次青年人才。主要从事量子物理与量子信息的基础理论研究，在国际一流专业刊物发表论文30余篇，其中Phys. Rev. Lett. 论文7篇、Nat. Commun.论文1篇。

邀请人

王东升 副研究员