Error exponent analysis in quantum information theory

University of Technology Sydney. Faculty of Engineering and Information Technology.Error exponent analysis aims at evaluating the exponential behaviour of the performance of the underlying system given a certain fixed coding rate. It is arguably a significant research topic in information theory because the analysis characterizes the trade-offs between the error probability of an information task, the size of the coding scheme, and the coding rate that determines the efficiency of the task. In this thesis, we give an exposition of error exponent analysis to two important quantum information processing protocols––classical data compression with quantum side information, and classical communications over quantum channels. We first prove substantial properties of various exponent functions, which allow us to better characterize the error behaviours of the tasks. Second, we establish accurate achievability and optimality finite blocklength bounds for the optimal error probability, providing useful and measurable benchmarks for future quantum information technology design. Finally, we extend the error exponent analysis to a more general setting where the coding rate is not fixed anymore, a research topic known as moderate deviation analysis. In other words, we show that the data recovery can be reliable when the compression rate approaches the conditional entropy slowly, and the reliable communication over a classical-quantum channel is possible as the transmission rate approaches channel capacity slowly. This line of research lies in the intersection of statistical analysis, matrix analysis, and information theory. Thus, the techniques employed in this studies could potentially be applicable to various areas such as classical and quantum information community, detection and estimation theory, statistics, and secrecy

A Simple and Tighter Derivation of Achievability for Classical Communication over Quantum Channels

Achievability in information theory refers to demonstrating a coding strategy that accomplishes a prescribed performance benchmark for the underlying task. In quantum information theory, the crafted Hayashi-Nagaoka operator inequality is an essential technique in proving a wealth of one-shot achievability bounds since it effectively resembles a union bound in various problems. In this work, we show that the pretty-good measurement naturally plays a role as the union bound as well. A judicious application of it considerably simplifies the derivation of one-shot achievability for classical-quantum (c-q) channel coding via an elegant three-line proof. The proposed analysis enjoys the following favorable features: (i) The established one-shot bound admits a closed-form expression as in the celebrated Holevo-Helstrom Theorem. Namely, the average error probability of sending $M$ messages through a c-q channel is upper bounded by the error of distinguishing the joint state between channel input and output against $(M-1)$-many products of its marginals. (ii) Our bound directly yields asymptotic results in the large deviation, small deviation, and moderate deviation regimes in a unified manner. (iii) The coefficients incurred in applying the Hayashi-Nagaoka operator inequality are no longer needed. Hence, the derived one-shot bound sharpens existing results that rely on the Hayashi-Nagaoka operator inequality. In particular, we obtain the tightest achievable $\epsilon$-one-shot capacity for c-q channel heretofore, and it improves the third-order coding rate in the asymptotic scenario. (iv) Our result holds for infinite-dimensional Hilbert space. (v) The proposed method applies to deriving one-shot bounds for data compression with quantum side information, entanglement-assisted classical communication over quantum channels, and various quantum network information-processing protocols

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