We show that the probability distribution of the error exponent in i.i.d. code ensembles over classical-quantum (CQ) channels with arbitrary output states accumulates above a threshold that is strictly larger than the CQ random coding exponent (RCE) at low rates, while coinciding with it at rates close to the mutual information of the channel. This result, combined with the work by Dalai [1] and the recent ones by Renes [2] and Li and Yang [3], implies that the ensemble distribution of error exponents concentrates around the CQ RCE in the high rate regime. Moreover, in the same rate regime the threshold we derive coincides with the ensemble-average of the exponent, that is, the typical random coding (TRC) exponent [4].
We show that the probability distribution of the error exponent in i.i.d. code ensembles over classical-quantum (CQ) channels with arbitrary output states accumulates above a threshold that is strictly larger than the CQ random coding exponent (RCE) at low rates, while coinciding with it at rates close to the mutual information of the channel. This result, combined with the work by Dalai [1] and the recent ones by Renes [2] and Li and Yang [3], implies that the ensemble distribution of error exponents concentrates around the CQ RCE in the high rate regime. Moreover, in the same rate regime the threshold we derive coincides with the ensemble-average of the exponent, that is, the typical random coding (TRC) exponent [4].
We show two properties of i.i.d. code ensembles over classical-quantum (CQ) channels with arbitrary output states. The first property is that the probability distribution of the error exponent across the ensemble accumulates above a threshold that is strictly larger than the CQ random coding exponent (RCE) at low rates, while coinciding with it at rates close to the mutual information of the channel. This result, combined with the works by Dalai, Renes, Li and Yang and Cheng and Liu, implies that the ensemble distribution of error exponents concentrates around the CQ RCE in the high rate regime. Moreover, in the same rate region the threshold we derive coincides with the ensemble-average of the exponent, that is, the CQ typical random coding (TRC) exponent. The second property we derive is that the probability that a randomly selected code from an ensemble contains a nested code of the same rate such that each of its codewords achieves the CQ expurgated exponent goes to one asymptotically in the codeword length. Such nested code can be obtained by expurgating a vanishingly small fraction of codewords as the codeword length increases. This result refines Holevo's work [5], which proved that the expurgated exponent can be achieved by discarding the worst half of the codewords from at least one code in the ensemble.
A fundamental quantity of interest in Shannon theory, classical or quantum, is the error exponent of a given channel W and rate R: the constantE(W,R)which governs the exponential decay of decoding error when using ever larger optimal codes of fixed rate R to communicate over ever more (memoryless) instances of a given channel W. Nearly matching lower and upper bounds are well-known for classical channels. Here I show a lower bound on the error exponent of communication over arbitrary classical-quantum (CQ) channels which matches Dalai’s sphere-packing upper bound for rates above a critical value, exactly analogous to the case of classical channels. This proves a conjecture made by Holevo in his investigation of the problem. Unlike the classical case, however, the argument does not proceed via a refined analysis of a suitable decoder, but instead by leveraging a bound by Hayashi on the error exponent of the cryptographic task of privacy amplification. This bound is then related to the coding problem via tight entropic uncertainty relations and Gallager’s method of constructing capacity-achieving parity-check codes for arbitrary channels. Along the way, I find a lower bound on the error exponent of the task of compression of classical information relative to quantum side information that matches the sphere-packing upper bound of Cheng et al. In turn, the polynomial prefactors to the sphere-packing bound found by Cheng et al. may be translated to the privacy amplification problem, sharpening a recent result by Li, Yao, and Hayashi, at least for linear randomness extractors.
A fundamental quantity of interest in Shannon theory, classical or quantum, is the optimal error exponent of a given channel W and rate R: the constant E(W,R) which governs the exponential decay of decoding error when using ever larger codes of fixed rate R to communicate over ever more (memoryless) instances of a given channel W. Here I show that a bound by Hayashi [CMP 333, 335 (2015)] for an analogous quantity in privacy amplification implies a lower bound on the error exponent of communication over symmetric classical-quantum channels. The resulting bound matches Dalai’s [IEEE TIT 59, 8027 (2013)] sphere-packing upper bound for rates above a critical value, and reproduces the well-known classical result for symmetric channels. The argument proceeds by first relating the error exponent of privacy amplification to that of compression of classical information with quantum side information, which gives a lower bound that matches the sphere-packing upper bound of Cheng et al. [IEEE TIT 67, 902 (2021)]. In turn, the polynomial prefactors to the sphere-packing bound found by Cheng et al. may be translated to the privacy amplification problem, sharpening a recent result by Li, Yao, and Hayashi [arXiv:2111.01075 [quant-ph]], at least for linear randomness extractors.
The ensemble of regular low-definition parity-check (LDPC) codes is considered. Using concentration results on the weight distribution, lower bounds on the error rate of a random code in the ensemble are derived. These bounds hold with some confidence level. Combining these results with known lower bounds on the error exponent, confidence intervals on the error exponent, under maximum-likelihood (ML) decoding, are obtained. Over a large range of channel parameter and transmission rate values, when the graph connectivity is sufficiently large, the upper bound of the interval approaches the lower bound, and the probability that the error exponent is within the interval can be arbitrarily close to one. In fact, in this case the true error exponent approaches the maximum between the random coding and the expurgated random coding exponents, with probability that approaches one.
We introduce a new quantum decoder based on a variant of the pretty good measurement, but defined via an alternative matrix quotient. We then use this novel decoder to derive new lower bounds on the error exponent both in the one-shot and asymptotic regimes for the classical-quantum and the entanglement-assisted channel coding problems. Our bounds are expressed in terms of measured (for the one-shot bounds) and sandwiched (for the asymptotic bounds) channel Rényi mutual information of order between 1/2 and 1. The bounds are not comparable with some previously established bounds for general channels, yet they are tight (for rates close to capacity) when the channel is classical. Finally, we also use our new decoder to rederive Cheng’s recent tight bound on the decoding error probability, which implies that most existing asymptotic results also hold for the new decoder.
We study lower bounds on the optimal error probability in classical coding over classical-quantum channels at rates below the capacity, commonly termed quantum sphere-packing bounds. Winter and Dalai have derived such bounds for classical-quantum channels; however, the exponents in their bounds only coincide when the channel is classical. In this paper, we show that these two exponents admit a variational representation and are related by the Golden-Thompson inequality, reaffirming that Dalai's expression is stronger in general classical-quantum channels. Second, we establish a finite blocklength sphere-packing bound for classical-quantum channels, which significantly improves Dalai's prefactor from the order of subexponential to polynomial. Furthermore, the gap between the obtained error exponent for constant composition codes and the best known classical random coding exponent vanishes in the order of o(logn/n), indicating our sphere-packing bound is almost exact in the high rate regime. Finally, for a special class of symmetric classical-quantum channels, we can completely characterize its optimal error probability without the constant composition code assumption. The main technical contributions are two converse Hoeffding bounds for quantum hypothesis testing and the saddle-point properties of error exponent functions.