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arXiv:0904.1073 · IEEE Transactions on Communications

Performance of quantum data transmission systems in the presence of thermal noise

Gianfranco Cariolaro, Gianfranco L. Pierobon

Published Feb 1, 2010• 60 citations
arXivHTMLPDF

Abstract

In the literature the performance of quantum data transmission systems is usually evaluated in the absence of thermal noise. A more realistic approach taking into account the thermal noise is intrinsically more difficult because it requires dealing with Glauber coherent states in an infinite-dimensional space. In particular, the exact evaluation of the optimal measurement operators is a very difficult task, and numerical approximation is unavoidable. The paper faces the problem by approximating the P-representation of the noisy quantum states with a large but finite numbers of terms and applying to them the square root measurement (SRM) approach. Comparisons with cases where the exact solution are known show that the SRM approach gives quite accurate results. As application, the performance of quadrature amplitude modulation (QAM) and phase shift keying (PSK) systems is considered. In spite of the fact that the SRM approach is not optimal and overestimates the error probability, also in these cases the quantum detection maintains its superiority with respect to the classical homodyne detection.

DOI
10.1109/tcomm.2010.02.080013

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In the literature the performance of quantum data transmission systems is usually evaluated in the absence of thermal noise. A more realistic approach taking into account the thermal noise is intrinsically more difficult because it requires dealing with Glauber coherent states in an infinite-dimensional space. In particular, the exact evaluation of the optimal measurement operators is a very difficult task, and numerical approximation is unavoidable. The paper faces the problem by approximating the P-representation of the noisy quantum states with a large but finite numbers of terms and applying to them the square root measurement (SRM) approach. Comparisons with cases where the exact solution are known show that the SRM approach gives quite accurate results. As application, the performance of quadrature amplitude modulation (QAM) and phase shift keying (PSK) systems is considered. In spite of the fact that the SRM approach is not optimal and overestimates the error probability, also in these cases the quantum detection maintains its superiority with respect to the classical homodyne detection.
The paper deals with quantum pulse position modulation (PPM), both in the absence (pure states) and in the presence (mixed states) of thermal noise, using the Glauber representation of coherent laser radiation. The objective is to find optimal (or suboptimal) measurement operators and to evaluate the corresponding error probability. For PPM, the correct formulation of quantum states is given by the tensorial product of m identical Hilbert spaces, where m is the PPM order. The presence of mixed states, due to thermal noise, generates an optimization problem involving matrices of huge dimensions, which already for 4-PPM, are of the order of ten thousand. To overcome this computational complexity, the currently available methods of quantum detection, which are based on explicit results, convex linear programming and square root measurement, are compared to find the computationally less expensive one. In this paper a fundamental role is played by the geometrically uniform symmetry of the quantum PPM format. The evaluation of error probability confirms the vast superiority of the quantum detection over its classical counterpart.
In quantum discrimination, the value of the minimum error probability and the set of measurement operators which achieve this minimum are often difficult to derive. Here we present a comparison of the performance obtained by the optimal solution and by the available bounds, namely the square root measurement (SRM) and the Chernoff bound. Applied to some Gaussian states, namely to coherent states with thermal noise, it is shown that the SRM provides a much tighter bound with respect to the Chernoff bound, with a comparable numerical complexity.
We analyze the performance of the error probability and the mutual information for quadrature amplitude modulation (QAM) and phase-shift keying systems based on quantum detection theory. It is shown that the quantum receiver called square root measurement gives about 5.7 dB improvement in power in comparison with the classical one. Furthermore, we show that the quantum QAM system can achieve the same reliability as the conventional intensity modulation-direct detection (IM-DD) system with almost equal power, compressing the required bandwidth, while the reliability is degraded in general.
A new technique for evaluating the performance of quantum signals observed in the presence of noise is described and evaluated. The quantum theory for detecting coherent-state signals has been developed previously; however, the quantum signal-plus-noise problem has received little attention due to its complexity. Here we develop a discrete approximation to the coherent-state representation of signal-plusnoise density operators, and present solutions to optimum receiver performance in terms of quantum measurement states whose performance is optimized via generalized rotations in Hilbert space. An efficient algorithm for carrying out the required numerical optimization is described and applied to binary signals observed in the presence of noise, for which exact results are available for comparison. The algorithm is then applied to the detection of ternary signals observed in the presence of noise, a previously unsolved problem, and the performance of the optimum receiver is characterized.
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.