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Allerton Conference on Communication, Control, and Computing

Superadditivity of quantum channel coding rate with finite blocklength quantum measurements

Hye Won Chung, Lizhong Zheng

Published Oct 1, 2013• 2 citations
Publisher

Abstract

This paper investigates the superadditivity of the maximum achievable information rate in a quantum channel. The superadditivity is a phenomenon that the maximum accessible information per channel strictly increases as the number of quantum states jointly measured increases. We give an explanation of this phenomenon by comparing the quantum channel with the classical discrete memoryless channel (DMC) under the concatenated codes, and give a lower bound on the maximum accessible information per channel at a finite length of quantum measurements in terms of V , which is the quantum version of channel dispersion, and C, the capacity of the quantum channel.

DOI
10.1109/allerton.2013.6736608

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This paper investigates the superadditivity of the maximum achievable information rate in a quantum channel. The superadditivity is a phenomenon that the maximum accessible information per channel strictly increases as the number of quantum states jointly measured increases. We give an explanation of this phenomenon by comparing the quantum channel with the classical discrete memoryless channel (DMC) under the concatenated codes, and give a lower bound on the maximum accessible information per channel at a finite length of quantum measurements in terms of V , which is the quantum version of channel dispersion, and C, the capacity of the quantum channel.
We investigate superadditivity in the maximum achievable rate of reliable classical communication over a quantum channel. The maximum number of classical information bits extracted per use of the quantum channel strictly increases as the number of channel outputs jointly measured at the receiver increases. This phenomenon is called superadditivity. We provide an explanation of this phenomenon by comparing a quantum channel with a classical discrete memoryless channel (DMC) under concatenated codes. We also give a lower bound on the maximum accessible information per channel use at a finite length of quantum measurements in terms of V, which is the quantum version of channel dispersion, and C, the classical capacity of the quantum channel.
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 CN/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 CN/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.
The entanglement-assisted capacity of a quantum channel admits an additive single-letter characterization, implying that joint encodings across channel uses cannot increase the ultimate communication rate. Here, we show that this additive picture does not extend to communication reliability. Specifically, we prove that the Petz-Rényi channel information can be strictly superadditive for every α∈[1/2,1)α\in[1/2,1)α∈[1/2,1), yielding a genuine multi-copy enhancement of the entanglement-assisted random-coding error exponent, even though the entanglement-assisted capacity remains additive. We establish this phenomenon analytically already for measurement channels, which are entanglement-breaking and have additive unassisted capacity. Remarkably, this strict superadditivity is witnessed by a separable, classically correlated two-copy channel-input marginal, demonstrating that no entanglement between the transmitted systems is required. Our results show that, although correlations across channel uses cannot increase the ultimate rate of entanglement-assisted communication, they can enhance its reliability.
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.
Quantum channel capacity is a fundamental quantity in order to understand how well quantum information can be transmitted or corrected when subjected to noise. However, it is generally not known how to compute such quantities since the quantum channel coherent information is not additive for all channels, implying that it must be maximized over an unbounded number of channel uses. This leads to the phenomenon known as superadditivity, which refers to the fact that the regularized coherent information of nnn channel uses exceeds one-shot coherent information. In this article, we study how the gain in quantum capacity of qudit depolarizing channels relates to the dimension of the considered systems. We make use of an argument based on the no-cloning bound in order to prove that the possible superadditive effects decrease as a function of the dimension for such family of channels. In addition, we prove that the capacity of the qudit depolarizing channel coincides with the coherent information when $d\ensuremath{\rightarrow}\ensuremath{\infty}$. We also discuss the private classical capacity and obtain similar results. We conclude that when high-dimensional qudits experiencing depolarizing noise are considered, the coherent information of the channel is not only an achievable rate, but essentially the maximum possible rate for any quantum block code.
We derive upper bounds on the rate of transmission of classical information over quantum channels by block codes with a given blocklength and error probability, for both entanglement-assisted and unassisted codes, in terms of a unifying framework of quantum hypothesis testing with restricted measurements. Our bounds do not depend on any special property of the channel (such as memorylessness) and generalize both a classical converse of Polyanskiy, Poor, and Verdú as well as a quantum converse of Renner and Wang, and have a number of desirable properties. In particular, our bound on entanglement-assisted codes is a semidefinite program and for memoryless channels, its large blocklength limit is the well-known formula for entanglement-assisted capacity due to Bennett, Shor, Smolin, and Thapliyal.