2,372 research outputs found
Constant Compositions in the Sphere Packing Bound for Classical-Quantum Channels
The sphere packing bound, in the form given by Shannon, Gallager and
Berlekamp, was recently extended to classical-quantum channels, and it was
shown that this creates a natural setting for combining probabilistic
approaches with some combinatorial ones such as the Lov\'asz theta function. In
this paper, we extend the study to the case of constant composition codes. We
first extend the sphere packing bound for classical-quantum channels to this
case, and we then show that the obtained result is related to a variation of
the Lov\'asz theta function studied by Marton. We then propose a further
extension to the case of varying channels and codewords with a constant
conditional composition given a particular sequence. This extension is then
applied to auxiliary channels to deduce a bound which can be interpreted as an
extension of the Elias bound.Comment: ISIT 2014. Two issues that were left open in Section IV of the first
version are now solve
A Simple Derivation of the Refined Sphere Packing Bound Under Certain Symmetry Hypotheses
A judicious application of the Berry-Esseen theorem via suitable Augustin
information measures is demonstrated to be sufficient for deriving the sphere
packing bound with a prefactor that is
for all codes on certain
families of channels -- including the Gaussian channels and the non-stationary
Renyi symmetric channels -- and for the constant composition codes on
stationary memoryless channels. The resulting non-asymptotic bounds have
definite approximation error terms. As a preliminary result that might be of
interest on its own, the trade-off between type I and type II error
probabilities in the hypothesis testing problem with (possibly non-stationary)
independent samples is determined up to some multiplicative constants, assuming
that the probabilities of both types of error are decaying exponentially with
the number of samples, using the Berry-Esseen theorem.Comment: 20 page
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
Lower bounds on the Probability of Error for Classical and Classical-Quantum Channels
In this paper, lower bounds on error probability in coding for discrete
classical and classical-quantum channels are studied. The contribution of the
paper goes in two main directions: i) extending classical bounds of Shannon,
Gallager and Berlekamp to classical-quantum channels, and ii) proposing a new
framework for lower bounding the probability of error of channels with a
zero-error capacity in the low rate region. The relation between these two
problems is revealed by showing that Lov\'asz' bound on zero-error capacity
emerges as a natural consequence of the sphere packing bound once we move to
the more general context of classical-quantum channels. A variation of
Lov\'asz' bound is then derived to lower bound the probability of error in the
low rate region by means of auxiliary channels. As a result of this study,
connections between the Lov\'asz theta function, the expurgated bound of
Gallager, the cutoff rate of a classical channel and the sphere packing bound
for classical-quantum channels are established.Comment: Updated to published version + bug fixed in Figure
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
Quantum Sphere-Packing Bounds with Polynomial Prefactors
© 1963-2012 IEEE. We study lower bounds on the optimal error probability in classical coding over classical-quantum channels at rates below the capacity, commonly termed quantum sphere-packing bounds. Winter and Dalai have derived such bounds for classical-quantum channels; however, the exponents in their bounds only coincide when the channel is classical. In this paper, we show that these two exponents admit a variational representation and are related by the Golden-Thompson inequality, reaffirming that Dalai's expression is stronger in general classical-quantum channels. Second, we establish a finite blocklength sphere-packing bound for classical-quantum channels, which significantly improves Dalai's prefactor from the order of subexponential to polynomial. Furthermore, the gap between the obtained error exponent for constant composition codes and the best known classical random coding exponent vanishes in the order of , indicating our sphere-packing bound is almost exact in the high rate regime. Finally, for a special class of symmetric classical-quantum channels, we can completely characterize its optimal error probability without the constant composition code assumption. The main technical contributions are two converse Hoeffding bounds for quantum hypothesis testing and the saddle-point properties of error exponent functions
Elias Bound for General Distances and Stable Sets in Edge-Weighted Graphs
This paper presents an extension of the Elias bound on the minimum distance
of codes for discrete alphabets with general, possibly infinite-valued,
distances. The bound is obtained by combining a previous extension of the Elias
bound, introduced by Blahut, with an extension of a bound previously introduced
by the author which builds upon ideas of Gallager, Lov\'asz and Marton. The
result can in fact be interpreted as a unification of the Elias bound and of
Lov\'asz's bound on graph (or zero-error) capacity, both being recovered as
particular cases of the one presented here. Previous extensions of the Elias
bound by Berlekamp, Blahut and Piret are shown to be included as particular
cases of our bound. Applications to the reliability function are then
discussed.Comment: Accepted, IEEE Transaction on Information Theor
Refined Strong Converse for the Constant Composition Codes
A strong converse bound for constant composition codes of the form
is
established using the Berry-Esseen theorem through the concepts of Augustin
information and Augustin mean, where is a constant determined by the
channel , the composition , and the rate , i.e., does not depend
on the block length .Comment: 7 page
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
Moderate deviation analysis for classical-quantum channels and quantum hypothesis testing
© 1963-2012 IEEE. In this paper, we study the tradeoffs between the error probabilities of classical-quantum channels and the blocklength n when the transmission rates approach the channel capacity at a rate lower than 1 {n} , a research topic known as moderate deviation analysis. We show that the optimal error probability vanishes under this rate convergence. Our main technical contributions are a tight quantum sphere-packing bound, obtained via Chaganty and Sethuraman's concentration inequality in strong large deviation theory, and asymptotic expansions of error-exponent functions. Moderate deviation analysis for quantum hypothesis testing is also established. The converse directly follows from our channel coding result, while the achievability relies on a martingale inequality
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