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Entropy

Simplification of the Gram Matrix Eigenvalue Problem for Quadrature Amplitude Modulation Signals

Ryusuke Miyazaki, Tiancheng Wang, Tsuyoshi Sasaki Usuda

Published Apr 13, 2022• 5 citations
Publisher

Abstract

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.

DOI
10.3390/e24040544

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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.
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.
The quadrature amplitude modulation (QAM) signal of coherent state of light is applied to the quantum stream cipher by Y-00 protocol. We first discuss on the performance of the square-root measurement (SRM) for the QAM signals in comparison with the optimum receiver. It is shown that the quantum stream cipher with the QAM signals is designed by using the SRM, taking account of the ciphertext-only attack and the known/chosen plain attack. Furthermore, the modification of the quantum stream cipher with the QAM signals is considered.
In quantum communication theory, group covariant signals are important and useful when we consider the minimization problem of the error probability or the maximization problem of the mutual information. However, in practical applications, there are important non-symmetric signals such as amplitude shift keying or quadrature amplitude modulation signals. To manage this situation, we have proposed a symmetrization method of non-symmetric pure-state signals by using coding. In this paper, we consider to apply the symmetrization method to mixed-state signals and give a necessary condition of mixed-state signals to apply the method. Then, we present examples of mixed-state signals to which the symmetrization method can be applied.
The square-root measurement (SRM) is regarded as the optimum quantum measurement for symmetric signals and quasi-optimum measurements for any quantum signals. Therefore, deriving formulae for the channel matrix by using SRM is crucial in developing a theory of quantum communications. To date, formulae for these matrices are known for symmetric and group-covariant signals. However, there are important non-symmetric signals such as ASK, AMPM, and QAM signals, etc. In this paper, we demonstrate that the calculation of the channel matrix for 4m-ary AMPM signals can be simplified by using their partial symmetry.
A unified treatment of typical optical modulation formats is given toward a future development of the quantum enigma cipher. In our analysis, the intensity shift keying (ISK), amplitude shift keying (ASK), phase shift keying (PSK), and quadrature amplitude modulation (QAM) coherent state signals are compared in terms of the masking number of potential signals through the numerical calculations of the square-root measurement. Through this comparison, we see that the use of ISK coherent state signal is the best choice for cryptographic purpose among these formats in achieving an appropriate balance between the number of signal photons and the number of quantum state signals.
Applying the quadrature amplitude modulation (QAM) format, quantum-noise randomized cipher (QNRC) systems hide the signal states in quantum phase noise and amplitude noise to prevent eavesdropping. In this paper, based on the traditional wire-tap channel model analysis method, the physical-layer security of QAM-QNRC system is investigated quantitatively under the metric of secrecy rate. The general expressions of secrecy rates of the data and key are derived separately. Furthermore, the maximum reachable secrecy rate of a QAM-QNRC system is put forward, under which the data and key are both safe in the view of mutual information evaluation. Finally, the variation trend of secrecy rate with various system parameters is discussed in detail. The simulation results show that we can obtain a higher secrecy rate by setting reasonable parameters, such as the level of ciphertext, mesoscopic signal power, and inner gain at the transmitter. Meanwhile, the security of the key is the main constraint of the maximum reachable secrecy rate.