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arXiv:quant-ph/9706061 · Physical Review A

Quantum-channel capacity of very noisy channels

David Peter Divincenzo, Peter W. Shor, John A. Smolin

Published Feb 1, 1998• 297 citations
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Abstract

We present a family of additive quantum error-correcting codes whose capacities exceed those of quantum random coding (hashing) for very noisy channels. These codes provide nonzero capacity in a depolarizing channel for fidelity parameters $f$ when $f>0.80944$. Random coding has nonzero capacity only for $f>0.81071$; by analogy to the classical Shannon coding limit, this value had previously been conjectured to be a lower bound. We use the method introduced by Shor and Smolin of concatenating a nonrandom repetition (cat) code within a random code to obtain good codes. The cat code with block size five is shown to be optimal for single concatenation. The best known multiple-concatenated code we found has a block size of 25. We derive a general relation between the capacity attainable by these concatenation schemes and the coherent information of the inner code states.

DOI
10.1103/physreva.57.830

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We present a family of additive quantum error-correcting codes whose capacities exceed those of quantum random coding (hashing) for very noisy channels. These codes provide nonzero capacity in a depolarizing channel for fidelity parameters fff when f>0.80944f>0.80944f>0.80944. Random coding has nonzero capacity only for f>0.81071f>0.81071f>0.81071; by analogy to the classical Shannon coding limit, this value had previously been conjectured to be a lower bound. We use the method introduced by Shor and Smolin of concatenating a nonrandom repetition (cat) code within a random code to obtain good codes. The cat code with block size five is shown to be optimal for single concatenation. The best known multiple-concatenated code we found has a block size of 25. We derive a general relation between the capacity attainable by these concatenation schemes and the coherent information of the inner code states.
We work out a theory of approximate quantum error correction that allows us to derive a general lower bound for the entanglement fidelity of a quantum code. The lower bound is given in terms of Kraus operators of the quantum noise. This result is then used to analyze the average error correcting performance of codes that are randomly drawn from unitarily invariant code ensembles. Our results confirm that random codes of sufficiently large block size are highly suitable for quantum error correction. Moreover, employing a lemma of Bennett, Shor, Smolin, and Thapliyal, we prove that random coding attains information rates of the regularized coherent information.
Abstract The classical noisy-channel coding theorem developed by Shannon establishes a fundamental limit on reliable information transmission. In quantum regime, this framework is extended by results such as the Lloyd-Shor-Devetak theorem, which gives the hashing limit on quantum channel capacity. In this work, we beat the hashing limit for low rate coding schemes over quantum Pauli channels. Our approach is based on a re-examination of Bacon-Shor codes and their generalizations, which we show are not merely simple constructions but instances of a broader mechanism that fundamentally alters achievable rate limits. Over depolarizing channels with depolarizing probability of p, these codes exceed the hashing bound with 0.1145 ≤ p ≤ 0.75, and further surpass the asymptotic quantum Singleton bound under bounded minimum-distance decoding for 0.3301 ≤ p ≤ 0.75. We further present explicit constructions based on quantum low-density parity-check codes, achieving frame error rates below 10−4 while exceeding the hashing bound. The advantage persists and is pronouncedly enhanced in the presence of biased noise, demonstrating robustness across realistic quantum Pauli channels. Our results challenge the prevailing understanding of fundamental limits in quantum coding and information theory.
We study encodings that give the best-known thresholds for the nonzero capacity of quantum channels, i.e., the upper bound for correctable noise, using an entropic approach to calculation of the threshold values. Our results show that Pauli noise is correctable up to the hashing bound. For a depolarizing channel, this approach allows one to achieve a nonzero capacity for a fidelity (probability of no error) of f=0.80870f=0.808\phantom{\rule{0.2em}{0ex}}70f=0.80870.
This thesis provides bounds on the performance of quantum error correcting codes when used for quantum communication and quantum key distribution. The first two chapters provide a bare-bones introduction to classical and quantum error correcting codes, respectively. The next four chapters present achievable rates for quantum codes in various scenarios. The final chapter is dedicated to an upper bound on the quantum channel capacity. Chapter 3 studies coding for adversarial noise using quantum list codes, showing there exist quantum codes with high rates and short lists. These can be used, together with a very short secret key, to communicate with high fidelity at noise levels for which perfect fidelity is, impossible. Chapter 4 explores the performance of a family of degenerate codes when used to communicate over Pauli channels, showing they can be used to communicate over almost any Pauli channel at rates that are impossible for a nondegenerate code and that exceed those of previously known degenerate codes. By studying the scaling of the optimal block length as a function of the channel's parameters, we develop a heuristic for designing even better codes. Chapter 5 describes an equivalence between a family of noisy preprocessing protocols for quantum key distribution and entanglement distillation protocols whose target state belongs to a class of private states called twisted states. In Chapter 6, the codes of Chapter 4 are combined with the protocols of Chapter 5 to provide higher key rates for one-way quantum key distribution than were previously thought possible. Finally, Chapter 7 presents a new upper bound on the quantum channel that is both additive and convex, and which can be interpreted as the of the channel for communication given access to side channels from a class of zero cloning channels. This clone assisted capacity is equal to the unassisted for channels that are degradable, which we use to find new upper bounds on the of a depolarizing channel.
An upper limit is given to the amount of quantum information that can be transmitted reliably down a noisy, decoherent quantum channel using the high-probability states of quantum sources. A class of quantum error-correcting codes is presented that allows the information transmitted to attain this limit. The result is a quantum analog of Shannon's bound and code for the noisy classical channel [C. E. Shannon and W. Weaver, The Mathematical Theory of Communication (University of Illinois Press, Chicago, 1948)].