Finite-Block-Length Analysis in Classical and Quantum Information Theory

Coding technology is used in several information processing tasks. In particular, when noise during transmission disturbs communications, coding technology is employed to protect the information. However, there are two types of coding technology: coding in classical information theory and coding in quantum information theory. Although the physical media used to transmit information ultimately obey quantum mechanics, we need to choose the type of coding depending on the kind of information device, classical or quantum, that is being used. In both branches of information theory, there are many elegant theoretical results under the ideal assumption that an infinitely large system is available. In a realistic situation, we need to account for finite size effects. The present paper reviews finite size effects in classical and quantum information theory with respect to various topics, including applied aspects

Universal Secure Multiplex Network Coding with Dependent and Non-Uniform Messages

We consider the random linear precoder at the source node as a secure network coding. We prove that it is strongly secure in the sense of Harada and Yamamoto and universal secure in the sense of Silva and Kschischang, while allowing arbitrary small but nonzero mutual information to the eavesdropper. Our security proof allows statistically dependent and non-uniform multiple secret messages, while all previous constructions of weakly or strongly secure network coding assumed independent and uniform messages, which are difficult to be ensured in practice.Comment: 10 pages, 1 figure, IEEEtrans.cls. Online published in IEEE Trans. Inform. Theor

Universal quantum information compression and degrees of prior knowledge

We describe a universal information compression scheme that compresses any pure quantum i.i.d. source asymptotically to its von Neumann entropy, with no prior knowledge of the structure of the source. We introduce a diagonalisation procedure that enables any classical compression algorithm to be utilised in a quantum context. Our scheme is then based on the corresponding quantum translation of the classical Lempel-Ziv algorithm. Our methods lead to a conceptually simple way of estimating the entropy of a source in terms of the measurement of an associated length parameter while maintaining high fidelity for long blocks. As a by-product we also estimate the eigenbasis of the source. Since our scheme is based on the Lempel-Ziv method, it can be applied also to target sequences that are not i.i.d.Comment: 17 pages, no figures. A preliminary version of this work was presented at EQIS '02, Tokyo, September 200

Exponents of quantum fixed-length pure state source coding

We derive the optimal exponent of the error probability of the quantum fixed-length pure state source coding in both cases of blind coding and visible coding. The optimal exponent is universally attained by Jozsa et al. (PRL, 81, 1714 (1998))'s universal code. In the direct part, a group representation theoretical type method is essential. In the converse part, Nielsen and Kempe (PRL, 86, 5184 (2001))'s lemma is essential.Comment: LaTeX2e and revetx4 with aps,twocolumn,superscriptaddress,showpacs,pra,amssymb,amsmath. The previous version has a mistak

On privacy amplification, lossy compression, and their duality to channel coding

We examine the task of privacy amplification from information-theoretic and coding-theoretic points of view. In the former, we give a one-shot characterization of the optimal rate of privacy amplification against classical adversaries in terms of the optimal type-II error in asymmetric hypothesis testing. This formulation can be easily computed to give finite-blocklength bounds and turns out to be equivalent to smooth min-entropy bounds by Renner and Wolf [Asiacrypt 2005] and Watanabe and Hayashi [ISIT 2013], as well as a bound in terms of the $E_\gamma$ divergence by Yang, Schaefer, and Poor [arXiv:1706.03866 [cs.IT]]. In the latter, we show that protocols for privacy amplification based on linear codes can be easily repurposed for channel simulation. Combined with known relations between channel simulation and lossy source coding, this implies that privacy amplification can be understood as a basic primitive for both channel simulation and lossy compression. Applied to symmetric channels or lossy compression settings, our construction leads to proto- cols of optimal rate in the asymptotic i.i.d. limit. Finally, appealing to the notion of channel duality recently detailed by us in [IEEE Trans. Info. Theory 64, 577 (2018)], we show that linear error-correcting codes for symmetric channels with quantum output can be transformed into linear lossy source coding schemes for classical variables arising from the dual channel. This explains a "curious duality" in these problems for the (self-dual) erasure channel observed by Martinian and Yedidia [Allerton 2003; arXiv:cs/0408008] and partly anticipates recent results on optimal lossy compression by polar and low-density generator matrix codes.Comment: v3: updated to include equivalence of the converse bound with smooth entropy formulations. v2: updated to include comparison with the one-shot bounds of arXiv:1706.03866. v1: 11 pages, 4 figure