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Electrical Engineering in Japan

Properties of quantum gain of coding with information criterion by binary linear codes

Yuki Ishida, Shogo Usami, Tsuyoshi Sasaki Usuda, Ichi Takumi

Published Apr 23, 2008• 3 citations
Publisher

Abstract

Abstract In quantum information theory, superadditivity in the capacity of a quantum channel occurs as a special property. We present a method of calculating the mutual information analytically for binary linear codes by using square‐root measurement as the decoding process. Many examples of codes showing the existence of superadditivity in capacity have been given in the past, but the scope of the method was not clarified. In the present paper, we show that the method can be applied to any binary linear codes. We also show that the quantum channel capacity is almost fully attained in a finite codeword length by using approximation to simplex code. © 2008 Wiley Periodicals, Inc. Electr Eng Jpn, 163(4): 48–57, 2008; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/eej.20646

DOI
10.1002/eej.20646

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Abstract In quantum information theory, superadditivity in the capacity of a quantum channel occurs as a special property. We present a method of calculating the mutual information analytically for binary linear codes by using square‐root measurement as the decoding process. Many examples of codes showing the existence of superadditivity in capacity have been given in the past, but the scope of the method was not clarified. In the present paper, we show that the method can be applied to any binary linear codes. We also show that the quantum channel capacity is almost fully attained in a finite codeword length by using approximation to simplex code. © 2008 Wiley Periodicals, Inc. Electr Eng Jpn, 163(4): 48–57, 2008; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/eej.20646
In quantum information theory, there is superadditivity in capacity of a quantum channel as special property. We showed the calculating method of mutual information analytically for binary linear codes by using Square-root measurement as decoding process. Therefore, many examples of codes which shows existence of superadditivity in capacity have been given. But the scope of method was not clarified. In the present paper, we show the method can be applied to any binary linear codes. Moreover, we show quantum channel capacity is almost attained in finite codeword length by using approximation to simplex code.
We consider a channel coding for sending classical information through a quantum channel with a given ensemble of quantum states (letter states). As is well known, it is generically possible in a quantum channel that the transmittable information in block coding of length nnn can exceed nnn times the maximum amount that can be sent without any coding scheme. This so-called superadditivity in classical capacity of a quantum channel is a distinct feature that cannot be found in a classical memoryless channel. In this paper, a practical model of channel coding that shows this property is presented. It consists of a simple code-word selection and the optimum decoding of the code words minimizing the average error probability. At first, optimization of decoding strategy is discussed. Then the channel coding that shows the superadditivity in classical capacity is demonstrated.
The polar codes proposed by Arikan have attracted much attention, because a channel capacity is achieved by the codes in the limit of the infinite codeword length by using successive cancellation decoding. Recently, a similar result was shown in the classical-quantum channel. In this paper, we calculate mutual information using optimum quantum decoding when the codeword length of the polar codes is finite and show that the quantum gain of coding is obtained using the polar codes. Finally, we compare the performance of the polar codes with that of other codes.
Quantum-information theory predicts that when the transmission resource is doubled in quantum channels, the amount of information transmitted can be increased more than twice by quantum-channel coding technique, whereas the increase is at most twice in classical information theory. This remarkable feature, the superadditive quantum-coding gain, can be implemented by appropriate choices of code words and corresponding quantum decoding which requires a collective quantum measurement. Recently, an experimental demonstration was reported [M. Fujiwara et al., Phys. Rev. Lett. 90, 167906 (2003)]. The purpose of this paper is to describe our experiment in detail. Particularly, a design strategy of quantum-collective decoding in physical quantum circuits is emphasized. We also address the practical implication of the gain on communication performance by introducing the quantum-classical hybrid coding scheme. We show how the superadditive quantum-coding gain, even in a small code length, can boost the communication performance of conventional coding techniques.
The maximum rate at which classical information can be reliably transmitted per use of a quantum channel strictly increases in general with N , the number of channel outputs that are detected jointly by the quantum joint-detection receiver (JDR). This phenomenon is known as superadditivity of the maximum achievable information rate over a quantum channel. We study this phenomenon for a pure-state classical-quantum channel and provide a lower bound on C[subscript N]/N, the maximum information rate when the JDR is restricted to making joint measurements over no more than N quantum channel outputs, while allowing arbitrary classical error correction. We also show the appearance of a superadditivity phenomenon-of mathematical resemblance to the aforesaid problem-in the channel capacity of a classical discrete memoryless channel when a concatenated coding scheme is employed, and the inner decoder is forced to make hard decisions on N-length inner codewords. Using this correspondence, we develop a unifying framework for the above two notions of superadditivity, and show that for our lower bound to C[subscript N]/N to be equal to a given fraction of the asymptotic capacity C of the respective channel, N must be proportional to V/C², where V is the respective channel dispersion quantity.