Lower bounds on the Probability of Error for Classical and Classical-Quantum Channels

In this paper, lower bounds on error probability in coding for discrete classical and classical-quantum channels are studied. The contribution of the paper goes in two main directions: i) extending classical bounds of Shannon, Gallager and Berlekamp to classical-quantum channels, and ii) proposing a new framework for lower bounding the probability of error of channels with a zero-error capacity in the low rate region. The relation between these two problems is revealed by showing that Lov\'asz' bound on zero-error capacity emerges as a natural consequence of the sphere packing bound once we move to the more general context of classical-quantum channels. A variation of Lov\'asz' bound is then derived to lower bound the probability of error in the low rate region by means of auxiliary channels. As a result of this study, connections between the Lov\'asz theta function, the expurgated bound of Gallager, the cutoff rate of a classical channel and the sphere packing bound for classical-quantum channels are established.Comment: Updated to published version + bug fixed in Figure

Quantum Sphere-Packing Bounds with Polynomial Prefactors

© 1963-2012 IEEE. 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 $o(\log n / n)$ , 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

Constant Compositions in the Sphere Packing Bound for Classical-Quantum Channels

The sphere packing bound, in the form given by Shannon, Gallager and Berlekamp, was recently extended to classical-quantum channels, and it was shown that this creates a natural setting for combining probabilistic approaches with some combinatorial ones such as the Lov\'asz theta function. In this paper, we extend the study to the case of constant composition codes. We first extend the sphere packing bound for classical-quantum channels to this case, and we then show that the obtained result is related to a variation of the Lov\'asz theta function studied by Marton. We then propose a further extension to the case of varying channels and codewords with a constant conditional composition given a particular sequence. This extension is then applied to auxiliary channels to deduce a bound which can be interpreted as an extension of the Elias bound.Comment: ISIT 2014. Two issues that were left open in Section IV of the first version are now solve

Sphere-packing bound for symmetric classical-quantum channels

© 2017 IEEE. "To be considered for the 2017 IEEE Jack Keil Wolf ISIT Student Paper Award." We provide a sphere-packing lower bound for the optimal error probability in finite blocklengths when coding over a symmetric classical-quantum channel. Our result shows that the pre-factor can be significantly improved from the order of the subexponential to the polynomial, This established pre-factor is arguably optimal because it matches the best known random coding upper bound in the classical case. Our approaches rely on a sharp concentration inequality in strong large deviation theory and crucial properties of the error-exponent function

Achievable error exponents of data compression with quantum side information and communication over symmetric classical-quantum channels

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.Comment: Comments very welcome

Sphere packing bound for quantum channels

In this paper, the Sphere-Packing-Bound of Fano, Shannon, Gallager and Berlekamp is extended to general classical-quantum channels. The obtained upper bound for the reliability function, for the case of pure-state channels, coincides at high rates with a lower bound derived by Burnashev and Holevo [1]. Thus, for pure state channels, the reliability function at high rates is now exactly determined. For the general case, the obtained upper bound expression at high rates was conjectured to represent also a lower bound to the reliability function, but a complete proof has not been obtained yet

Some remarks on classical and classical-quantum sphere packing bounds: Rényi vs. Kullback-Leibler

We review the use of binary hypothesis testing for the derivation of the sphere packing bound in channel coding, pointing out a key difference between the classical and the classical-quantum setting. In the first case, two ways of using the binary hypothesis testing are known, which lead to the same bound written in different analytical expressions. The first method historically compares output distributions induced by the codewords with an auxiliary fixed output distribution, and naturally leads to an expression using the Renyi divergence. The second method compares the given channel with an auxiliary one and leads to an expression using the Kullback-Leibler divergence. In the classical-quantum case, due to a fundamental difference in the quantum binary hypothesis testing, these two approaches lead to two different bounds, the first being the "right" one. We discuss the details of this phenomenon, which suggests the question of whether auxiliary channels are used in the optimal way in the second approach and whether recent results on the exact strong-converse exponent in classical-quantum channel coding might play a role in the considered proble

Moderate deviation analysis for classical-quantum channels and quantum hypothesis testing

© 1963-2012 IEEE. In this paper, we study the tradeoffs between the error probabilities of classical-quantum channels and the blocklength n when the transmission rates approach the channel capacity at a rate lower than 1 {n} , a research topic known as moderate deviation analysis. We show that the optimal error probability vanishes under this rate convergence. Our main technical contributions are a tight quantum sphere-packing bound, obtained via Chaganty and Sethuraman's concentration inequality in strong large deviation theory, and asymptotic expansions of error-exponent functions. Moderate deviation analysis for quantum hypothesis testing is also established. The converse directly follows from our channel coding result, while the achievability relies on a martingale inequality

Lov\'asz's Theta Function, R\'enyi's Divergence and the Sphere-Packing Bound

Lov\'asz's bound to the capacity of a graph and the the sphere-packing bound to the probability of error in channel coding are given a unified presentation as information radii of the Csisz\'ar type using the R{\'e}nyi divergence in the classical-quantum setting. This brings together two results in coding theory that are usually considered as being of a very different nature, one being a "combinatorial" result and the other being "probabilistic". In the context of quantum information theory, this difference disappears.Comment: An excerpt from arXiv:1201.5411v3 (with a different notation) accepted at ISIT 201

Properties of Noncommutative Renyi and Augustin Information

The scaled R\'enyi information plays a significant role in evaluating the performance of information processing tasks by virtue of its connection to the error exponent analysis. In quantum information theory, there are three generalizations of the classical R\'enyi divergence---the Petz's, sandwiched, and log-Euclidean versions, that possess meaningful operational interpretation. However, these scaled noncommutative R\'enyi informations are much less explored compared with their classical counterpart, and lacking crucial properties hinders applications of these quantities to refined performance analysis. The goal of this paper is thus to analyze fundamental properties of scaled R\'enyi information from a noncommutative measure-theoretic perspective. Firstly, we prove the uniform equicontinuity for all three quantum versions of R\'enyi information, hence it yields the joint continuity of these quantities in the orders and priors. Secondly, we establish the concavity in the region of $s\in(-1,0)$ for both Petz's and the sandwiched versions. This completes the open questions raised by Holevo [\href{https://ieeexplore.ieee.org/document/868501/}{\textit{IEEE Trans.~Inf.~Theory}, \textbf{46}(6):2256--2261, 2000}], Mosonyi and Ogawa [\href{https://doi.org/10.1007/s00220-017-2928-4/}{\textit{Commun.~Math.~Phys}, \textbf{355}(1):373--426, 2017}]. For the applications, we show that the strong converse exponent in classical-quantum channel coding satisfies a minimax identity. The established concavity is further employed to prove an entropic duality between classical data compression with quantum side information and classical-quantum channel coding, and a Fenchel duality in joint source-channel coding with quantum side information in the forthcoming papers