Simplification of the Gram Matrix Eigenvalue Problem for Quantum Signals Formed by Rotating Signal Points in a Circular Sector Region

Abstract

Efficient computation of eigenvalues and eigenvectors of the Gram matrix for quantum signals is desirable in the field of quantum communication theory. Because various quantities such as the error probability, mutual information, channel capacity, and upper and lower bounds of the reliability function can be obtained by the eigenvalues and eigenvectors. Moreover, 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 must be used. Recently, we have shown that, for certain asymmetric signals, the Gram matrix eigenvalue problem can be simplified and it is sufficient to consider the eigenvalue problem of smaller matrices at half or quarter size. In this paper, we consider nm-ary quantum signals formed by rotating signal points in a circular sector region and show that solving the 1/n-size matrix eigenvalue problem is sufficient to give the eigenvalues and eigenvectors of the Gram matrix.