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Science in China. Series G, Physics, mechanics & astronomy/Science in China. Series G, Physics, Mechanics & Astronomy

Classical information theory in quantum notations

Ye Ming

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

Shannon’s noisy channel coding theorem is a cornerstone in the history of classical information theory. A proof of Shannon’s noisy channel coding theorem using quantum notations is presented, which sets up a bridge between classical information theory and the developing quantum information theory.

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These notes (in French) provide a concise introduction to information theory, spanning its classical foundations: Shannon entropy, joint and conditional entropies, mutual information, statistical correlations, and Markov chains; as well as Shannon's source coding and channel capacity theorems, which govern the optimal compression and transmission of information through noisy channels. They also cover the quantum extensions of these concepts, including the density operator formalism, distances between quantum states, von Neumann entropy, coherent information, conditional entropy, entanglement theory, quantum discord, and quantum resource theory.
The more than thirty years old issue of the information capacity of quantum communication channels was dramatically clarified during when a number of direct quantum coding theorems were discovered. To a considerable extent this progress is due to an interplay between the quantum communication theory and quantum information ideas related to more recent development in quantum computing. It is remarkable, however, that many probabilistic tools underlying the treatment of the quantum case have their roots, and in some cases direct prototypes, in the classical Shannon theory.
Reverse Shannon theorems concern the use of noiseless channels to simulate noisy ones. This is dual to the usual noisy channel coding problem, where a noisy (classical or quantum) channel is used to simulate a noiseless one. The Quantum Reverse Shannon Theorem is extensively studied by Bennett and co-authors in [IEEE Trans. Inf. Theory, 2014]. They present two distinct theorems, each tailored to classical and quantum channel simulations respectively, explaining the fact that these theorems remain incomparable due to the fundamentally different nature of correlations they address. The authors leave as an open question the challenge of formulating a unified theorem that could encompass the principles of both and unify them. We unify these two theorems into a single, comprehensive theorem, extending it to the most general case by considering correlations with a general mixed-state reference system. Furthermore, we unify feedback and non-feedback theorems by simulating a general side information system at the encoder side.
An upper limit is given to the amount of quantum information that can be transmitted reliably down a noisy, decoherent quantum channel using the high-probability states of quantum sources. A class of quantum error-correcting codes is presented that allows the information transmitted to attain this limit. The result is a quantum analog of Shannon's bound and code for the noisy classical channel [C. E. Shannon and W. Weaver, The Mathematical Theory of Communication (University of Illinois Press, Chicago, 1948)].