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arXiv:0708.1597 · Physical Review A

Lower bounds on the nonzero capacity of Pauli channels

Jesse Fern, K. Birgitta Whaley

Published Dec 19, 2008• 70 citations
arXivHTMLPDF

Abstract

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 $f=0.808\phantom{\rule{0.2em}{0ex}}70$.

DOI
10.1103/physreva.78.062335

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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.
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.
The quantum capacity captures the value of a quantum channel for transmitting quantum information, establishing the fundamental limits on quantum communication. In spite of its central role in quantum information theory, the quantum capacity of most channels is unknown, with wide gaps between the best upper and lower bounds. Even deciding whether a channel has nonzero capacity -- finding its capacity threshold -- is difficult. In this paper we report significant increases in the capacity thresholds of two prototypical noise models: the depolarizing channel and Pauli channels. In the case of the depolarizing channel, this is the first improvement in 18 years, giving a bigger increase beyond the hashing bound than all previous improvements combined. Our starting point is the representation theoretic framework recently proposed by Bhalerao and Leditzky (2025) to compute coherent information for special permutation invariant states. We generalize their framework to the full symmetric subspace, which allow us to optimize coherent information over rank two states in that space. A representation theoretic calculation shows that exponentially many Kraus operators of the channel annihilate the symmetric space, corresponding to a massive decrease in environment entropy for states on the symmetric space compared to the maximally mixed state. This explains the enhanced coherent information as a manifestation of degeneracy for the resulting codes.
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On a class of quantum channels which includes the depolarizing channel, the highest fidelity of quantum error-correcting codes of length n and rate R is proven to be lower bounded by $1\ensuremath{-}\mathrm{exp}[\ensuremath{-}\mathrm{nE}(R)+o(n)]$ for some function E(R).E(R).E(R). The E(R)E(R)E(R) is positive below some threshold R0,{R}_{0},R0​, which implies R0{R}_{0}R0​ is a lower bound on the quantum capacity.
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 four well-known capacities of a two-parameter family of qubit Pauli channels. These are the channels that are covariant under the SO(2)\text{SO}(2)SO(2) group and contain the depolarizing channel as a special case. We find exact expressions for the classical capacity and entanglement-assisted capacities, and analytically determine the regions where the quantum capacity of the channel vanishes. We then use a flag extension to find upper bound for the quantum capacity and private capacity of these channels in the entire region of parameter space and also obtain the lower bound for the quantum capacity by calculating the single-shot quantum capacity numerically. In conjunction with previous results on depolarizing channels, our result is one step forward for determining the capacities of the full Pauli channel.