Miyazaki Ryusuke, Wang Tiancheng, Usuda Tsuyoshi Sasaki
Graduate School of Information Science and Technology, Aichi Prefectural University, Nagakute 480-1198, Aichi, Japan.
Faculty of Engineering, Kanagawa University, Yokohama 221-8686, Kanagawa, Japan.
Entropy (Basel). 2022 Apr 13;24(4):544. doi: 10.3390/e24040544.
In quantum information science, it is very important to solve the eigenvalue problem of the Gram matrix for quantum signals. This allows various quantities to be calculated, such as the error probability, mutual information, channel capacity, and the upper and lower bounds of the reliability function. Solving the eigenvalue problem also provides a matrix representation of quantum signals, which is useful for simulating quantum systems. In the case of symmetric signals, analytic solutions to the eigenvalue problem of the Gram matrix have been obtained, and efficient computations are possible. However, for asymmetric signals, there is no analytic solution and universal numerical algorithms that must be used, rendering the computations inefficient. Recently, we have shown that, for asymmetric signals such as amplitude-shift keying coherent-state signals, the Gram matrix eigenvalue problem can be simplified by exploiting its partial symmetry. In this paper, we clarify a method for simplifying the eigenvalue problem of the Gram matrix for quadrature amplitude modulation (QAM) signals, which are extremely important for applications in quantum communication and quantum ciphers. The results presented in this paper are applicable to ordinary QAM signals as well as modified QAM signals, which enhance the security of quantum cryptography.
在量子信息科学中，求解量子信号的格拉姆矩阵的特征值问题非常重要。这使得可以计算各种量，如错误概率、互信息、信道容量以及可靠性函数的上下界。求解特征值问题还提供了量子信号的矩阵表示，这对于模拟量子系统很有用。在对称信号的情况下，已经获得了格拉姆矩阵特征值问题的解析解，并且可以进行高效计算。然而，对于非对称信号，没有解析解且必须使用通用数值算法，这使得计算效率低下。最近，我们已经表明，对于诸如幅度键控相干态信号之类的非对称信号，可以通过利用其部分对称性来简化格拉姆矩阵特征值问题。在本文中，我们阐明了一种简化正交幅度调制（QAM）信号的格拉姆矩阵特征值问题的方法，QAM信号在量子通信和量子密码学应用中极其重要。本文给出的结果适用于普通QAM信号以及改进的QAM信号，改进的QAM信号增强了量子密码学的安全性。
