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arXiv:2203.09822 · IEEE International Symposium on Information Theory (ISIT)

Performance of Coherent Frequency-Shift Keying for Classical Communication on Quantum Channels

Matteo Rosati

Published Jul 12, 2021• 10 citations
arXivHTMLPDF

Abstract

We evaluate the performance of coherent frequency-shift keying (CFSK) [1], [2] alphabets for communication on quantum channels. We show that, contrarily to what previously thought, the square-root-measurement (SRM) is sub-optimal for discriminating CFSK states. Furthermore, we compute the maximum information transmission rate of the CFSK alphabet, observing that it employs at least as many frequency modes as the signal states, and compare it with standard phase-shift-keying. Finally, we introduce a discretized CFSK alphabet with higher mode-efficiency, exhibiting comparable error-probability performance with respect to CFSK and better rate performance. Our results suggest the existence of a tradeoff between the CFSK reduced error-probability and its mode efficiency.

DOI
10.1109/isit45174.2021.9517959

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We evaluate the performance of coherent frequency-shift keying (CFSK) [1], [2] alphabets for communication on quantum channels. We show that, contrarily to what previously thought, the square-root-measurement (SRM) is sub-optimal for discriminating CFSK states. Furthermore, we compute the maximum information transmission rate of the CFSK alphabet, observing that it employs at least as many frequency modes as the signal states, and compare it with standard phase-shift-keying. Finally, we introduce a discretized CFSK alphabet with higher mode-efficiency, exhibiting comparable error-probability performance with respect to CFSK and better rate performance. Our results suggest the existence of a tradeoff between the CFSK reduced error-probability and its mode efficiency.
The efficient use of communication channels motivates extensive research in novel communication protocols. Modern communication protocols use large alphabets contain up to a few thousand symbols, thus optimizing the use of power and available frequency space. To date, quantum receivers that discriminate up to approximately 20 symbols were theoretically investigated and receivers with as many as 4 nonorthogonal coherent states have been experimentally demonstrated. However, all heretofore explored quantum receivers suffer from the sensitivity degradation with the alphabet size. Particularly, their Helstrom Bound (HB) nearly reaches the classical standard quantum limit (SQL). Here we introduce an M-ary quantum receiver based on a coherent frequency shift keying (CFSK) protocol with a record power sensitivity, free of the above deficiency. The CFSK not only provides better accuracy for longer alphabets but also allows discrimination with a practically attainable symbol error rate (SER) situated much below the HBs of other encodings for large alphabets. Our receiver operates with a classical transmitter, and with any communication channel, including the existing global fiber network. It can be used to increase the amplification-free range in a network and/or reduce power requirements on the transmitter by more than 1000 times. In addition, the quantum measurement advantage can significantly optimize the use of the frequency space in comparison to classical frequency keying protocols. This advantage can be used in deep-space telecom links to enhance the satellite power budget. In existing fiber network links, quantum CFSK receivers can improve the amplification-free range by approximately the factor of 2.
In classical digital communications two main families of “M-ary” modulation schemes are generally distinguished: bandwidth limited and power limited. The canonical comparison of these modulation methods is based on normalized data rate (R/W) (bits per second per hertz of bandwidth) and the signal to noise rate per bit required to achieve a given error probability for different M. In a classical picture, the two families reside in two separate semi-plains R/W < 1 and R/W > 1, i.e. energy efficiency and bandwidth efficiency cannot be optimized at the same time. However, we find an alphabet family that can be paired with a quantum receiver to simultaneously optimize bandwidth and power efficiency of a communication channel. Particularly we found that coherent frequency shift keying (CFSK) gives rise to a family of communication protocols that are bandwidth limited in nature, but whose bandwidth usage can be optimized so that R/W>1 for a range of alphabet lengths M, while power sensitivity beats that of power-limited protocols. We will report our theoretical findings and experimental progress towards implementation of this protocol family.
Quantum mechanics allows measurements that surpass the fundamental sensitivity limits of classical methods. To benefit from the quantum advantage in a practical setting, the receiver should use communication channels resources optimally; this can be done employing large communication alphabets. Here we show the fundamental sensitivity potential of a quantum receiver for coherent communication with frequency shift keying. We introduce an adaptive quantum protocol for this receiver, show that its sensitivity outperforms other receivers for alphabet sizes above 4 and scales favorably, whereas quantum receivers explored to date suffer from degraded sensitivity with the alphabet size. In addition, we show that the quantum measurement advantage allows the much better use of the frequency space in comparison to classical frequency keying protocols and orthogonal frequency division multiplexing.
The optimal distributions on a channel input that yield the random coding exponent, the expurgation exponent, and the zero-rate reliability function for a discrete classical-quantum channel with M-ary Phase Shift Keying (PSK) coherent state signal are analytically derived. In each case, the optimal distributions are given by a uniform distribution over the channel input alphabet. As a result, the expressions of the error exponents for M-ary PSK coherent state signal are simplified. By using this result, typical behavior of the error exponents is sketched in the case of M = 16.
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.