4,322 research outputs found
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
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
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
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
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
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