Narrow-sense group covariant signals are important in applications such as phase-shift keying (PSK) coherent-state signals and coded symmetric signals. In addition, by using the character of a group, analytic solutions can be obtained for narrow-sense group covariant signals, which are a generalization of symmetric signals. However, if the signals are symmetric, the analytic solutions of the Gram matrix eigenvalues and eigenvectors can be obtained by using well-known operations in linear algebra. Therefore, if we use a universal numerical algorithm to compute the eigenvalues and eigenvectors of the Gram matrix, the computation is hard when M is large. In general, the Gram matrix is M × M for M-ary pure-state signals. Because the Gram matrix is a matrix representation of the density operator of the quantum information source, the Holevo capacity and the upper and lower bounds of the reliability function can be directly calculated by using its eigenvalues. Furthermore, even if the quantum state is a vector in an infinite-dimensional Hilbert space, such as a coherent state or squeezed state, the matrix form allows numerical calculations to be performed because it provides a representation in a finite-dimensional subspace (e.g., ). This representation is known to be useful for analyzing quantum systems (e.g., ). As each component of the square root of the Gram matrix corresponds to the inner product of a signal quantum state and a measurement state of the SRM, a matrix representation of the signal quantum state can be obtained when the signal quantum states are linearly independent. Actually, SRM is also strictly optimal for some asymmetric pure-state signals with not necessarily uniform a priori probabilities. Moreover, SRM is strictly optimal for symmetric pure-state signals with uniform a priori probabilities. SRM is asymptotically optimal for any quantum state signals with respect to minimizing the error probability, and it is used in the proof of the quantum channel coding theorem. This implies that the error probability and mutual information using SRM can be directly calculated. By solving the eigenvalue problem of the Gram matrix and finding its square root, the channel matrix given by the so-called square-root measurement (SRM) can be computed. The eigenvalues and eigenvectors of the Gram matrix are very useful for computing various quantities that evaluate system performance. However, recent experiments have shown that some cases may contain millions or even billions of signals. In quantum stream ciphers, the number of signals usually runs to several hundreds or thousands. In particular, because the security of a quantum stream cipher relies on the difference between the quantum optimum receiving capabilities of the legitimate receiver and the eavesdropper, it is essential to evaluate the optimum quantum receiver performance of the eavesdropper to guarantee security. The computation of these quantities is essential not only for evaluating the reliability of quantum communication and the sensitivity of quantum radar but also for guaranteeing the security of quantum cryptography. The efficient computations and evaluations of quantities such as the error probability, mutual information, channel capacity, and reliability function are extremely important in quantum communication, quantum radar, and quantum cipher systems. 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. 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. 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. However, for asymmetric signals, there is no analytic solution and universal numerical algorithms that must be used, rendering the computations inefficient. In the case of symmetric signals, analytic solutions to the eigenvalue problem of the Gram matrix have been obtained, and efficient computations are possible. Solving the eigenvalue problem also provides a matrix representation of quantum signals, which is useful for simulating quantum systems. 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. In quantum information science, it is very important to solve the eigenvalue problem of the Gram matrix for quantum signals.
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