Digital Object Identiﬁer (DOI) 10.1007/s00220-017-2971-1
Commun. Math. Phys. 355, 1283–1315 (2017)
Moderate Deviation Analysis for Classical Communication
over Quantum Channels
Christopher T. Chubb
, Vincent Y. F. Tan
, Marco Tomamichel
Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, Australia.
Department of Electrical and Computer Engineering, National University of Singapore,
Department of Mathematics, National University of Singapore, Singapore, Singapore.
Centre for Quantum Software and Information, University of Technology Sydney, Sydney, Australia.
Received: 15 February 2017 / Accepted: 11 June 2017
Published online: 2 August 2017 – © Springer-Verlag GmbH Germany 2017
Abstract: We analyse families of codes for classical data transmission over quantum
channels that have both a vanishing probability of error and a code rate approaching
capacity as the code length increases. To characterise the fundamental tradeoff between
decoding error, code rate and code length for such codes we introduce a quantum gen-
eralisation of the moderate deviation analysis proposed by Alt˘ug and Wagner as well
as Polyanskiy and Verdú. We derive such a tradeoff for classical-quantum (as well as
image-additive) channels in terms of the channel capacity and the channel dispersion,
giving further evidence that the latter quantity characterises the necessary backoff from
capacity when transmitting ﬁnite blocks of classical data. To derive these results we also
study asymmetric binary quantum hypothesis testing in the moderate deviations regime.
Due to the central importance of the latter task, we expect that our techniques will ﬁnd
further applications in the analysis of other quantum information processing tasks.
The goal of information theory is to ﬁnd the fundamental limits imposed on infor-
mation processing and transmission by the laws of physics. One of the early break-
throughs in quantum information theory was the characterisation of the capacity of a
classical-quantum (c-q) channel to transmit classical information by Holevo [1,2] and
Schumacher–Westmoreland . The classical capacity of a quantum channel is deﬁned
as the maximal rate (in bits per channel use) at which we can transmit information
such that the decoding error vanishes asymptotically as the length of the code increases.
However, for many practical applications there are natural restrictions on the code length
imposed, for example, by limitations on how much quantum information can be pro-
cessed coherently. Therefore it is crucial to go beyond the asymptotic treatment and
understand the intricate tradeoff between decoding error probability, code rate and code