基于矩阵填充技术重构低秩密度矩阵

Reconstructing low-rank density matrix via matrix completion

从有限的信息中重构低秩或者近似低秩矩阵的问题日益受到人们的关注,解决这个问题的技术称为矩阵填充.对于一个希尔伯特空间下纯态或者近似纯态的量子体系(也就是低熵状态),其密度矩阵是低秩的,且迹为1.将矩阵填充理论应用于重构泡利测量下未知密度矩阵中,用Matlab软件程序进行数值模拟,采用奇异值阈值算法,将软阈值法则用在未知态密度矩阵的奇异值上,通过计算机编程,进行阈值迭代,直至达到截止标准,能够大大提高运行速率.由于以泡利矩阵为基的张量积结构便于在实验中获得,以重构泡利测量下的未知密度矩阵为例,采集了部分数据,分析了矩阵的重构结果.通过对重构误差、运行时间、采样率方面的研究,得出密度矩阵能够通过矩阵填充技术完整重构的结论.

The problem of reconstructing low-rank or approximately low-rank matrix from the limited information is getting people's attention and solving this problem is well known as matrix completion. For the pure or nearly pure quantum state(ie. low entropy state) in a Hilbert space, the density matrix is low-rank and has trace 1. This paper is concerned with applying matrix completion theory to the unknown density matrix recovery which is from Pauli measurements. The singular value thresholding(SVT) algorithm was utilized to numerical simulation by Matlab software programs. And its soft-thresholding rule was used to singular values of the unknown state density matrix. The threshold iteration was conducted by SVT algorithm through computer programming until a stopping criteria was reached, which could greatly improve the run rate. We took the density matrix from Pauli measurements for example because of the convenience on getting the tensor product structure based on Pauli matrices in experiment. The effect of matrix reconstruction was studied in the case of sampling a small number of entries from the matrix. We conclude that the density matrix can be reconstructed successfully by studying the aspects of reconstruction error, run-time and sampling rate.