449 research outputs found
Second-Order Coding Rates for Channels with State
We study the performance limits of state-dependent discrete memoryless
channels with a discrete state available at both the encoder and the decoder.
We establish the epsilon-capacity as well as necessary and sufficient
conditions for the strong converse property for such channels when the sequence
of channel states is not necessarily stationary, memoryless or ergodic. We then
seek a finer characterization of these capacities in terms of second-order
coding rates. The general results are supplemented by several examples
including i.i.d. and Markov states and mixed channels
Ergodic Classical-Quantum Channels: Structure and Coding Theorems
We consider ergodic causal classical-quantum channels (cq-channels) which
additionally have a decaying input memory. In the first part we develop some
structural properties of ergodic cq-channels and provide equivalent conditions
for ergodicity. In the second part we prove the coding theorem with weak
converse for causal ergodic cq-channels with decaying input memory. Our proof
is based on the possibility to introduce joint input-output state for the
cq-channels and an application of the Shannon-McMillan theorem for ergodic
quantum states. In the last part of the paper it is shown how this result
implies coding theorem for the classical capacity of a class of causal ergodic
quantum channels.Comment: 19 pages, no figures. Final versio
Asymptotic Estimates in Information Theory with Non-Vanishing Error Probabilities
This monograph presents a unified treatment of single- and multi-user
problems in Shannon's information theory where we depart from the requirement
that the error probability decays asymptotically in the blocklength. Instead,
the error probabilities for various problems are bounded above by a
non-vanishing constant and the spotlight is shone on achievable coding rates as
functions of the growing blocklengths. This represents the study of asymptotic
estimates with non-vanishing error probabilities.
In Part I, after reviewing the fundamentals of information theory, we discuss
Strassen's seminal result for binary hypothesis testing where the type-I error
probability is non-vanishing and the rate of decay of the type-II error
probability with growing number of independent observations is characterized.
In Part II, we use this basic hypothesis testing result to develop second- and
sometimes, even third-order asymptotic expansions for point-to-point
communication. Finally in Part III, we consider network information theory
problems for which the second-order asymptotics are known. These problems
include some classes of channels with random state, the multiple-encoder
distributed lossless source coding (Slepian-Wolf) problem and special cases of
the Gaussian interference and multiple-access channels. Finally, we discuss
avenues for further research.Comment: Further comments welcom
The invalidity of a strong capacity for a quantum channel with memory
The strong capacity of a particular channel can be interpreted as a sharp
limit on the amount of information which can be transmitted reliably over that
channel. To evaluate the strong capacity of a particular channel one must prove
both the direct part of the channel coding theorem and the strong converse for
the channel. Here we consider the strong converse theorem for the periodic
quantum channel and show some rather surprising results. We first show that the
strong converse does not hold in general for this channel and therefore the
channel does not have a strong capacity. Instead, we find that there is a scale
of capacities corresponding to error probabilities between integer multiples of
the inverse of the periodicity of the channel. A similar scale also exists for
the random channel.Comment: 7 pages, double column. Comments welcome. Repeated equation removed
and one reference adde
General formulas for capacity of classical-quantum channels
The capacity of a classical-quantum channel (or in other words the classical
capacity of a quantum channel) is considered in the most general setting, where
no structural assumptions such as the stationary memoryless property are made
on a channel. A capacity formula as well as a characterization of the strong
converse property is given just in parallel with the corresponding classical
results of Verd\'{u}-Han which are based on the so-called information-spectrum
method. The general results are applied to the stationary memoryless case with
or without cost constraint on inputs, whereby a deep relation between the
channel coding theory and the hypothesis testing for two quantum states is
elucidated. no structural assumptions such as the stationary memoryless
property are made on a channel. A capacity formula as well as a
characterization of the strong converse property is given just in parallel with
the corresponding classical results of Verdu-Han which are based on the
so-called information-spectrum method. The general results are applied to the
stationary memoryless case with or without cost constraint on inputs, whereby a
deep relation between the channel coding theory and the hypothesis testing for
two quantum states is elucidated
Re-proving Channel Polarization Theorems: An Extremality and Robustness Analysis
The general subject considered in this thesis is a recently discovered coding
technique, polar coding, which is used to construct a class of error correction
codes with unique properties. In his ground-breaking work, Ar{\i}kan proved
that this class of codes, called polar codes, achieve the symmetric capacity
--- the mutual information evaluated at the uniform input distribution ---of
any stationary binary discrete memoryless channel with low complexity encoders
and decoders requiring in the order of operations in the
block-length . This discovery settled the long standing open problem left by
Shannon of finding low complexity codes achieving the channel capacity.
Polar coding settled an open problem in information theory, yet opened plenty
of challenging problems that need to be addressed. A significant part of this
thesis is dedicated to advancing the knowledge about this technique in two
directions. The first one provides a better understanding of polar coding by
generalizing some of the existing results and discussing their implications,
and the second one studies the robustness of the theory over communication
models introducing various forms of uncertainty or variations into the
probabilistic model of the channel.Comment: Preview of my PhD Thesis, EPFL, Lausanne, 2014. For the full version,
see http://people.epfl.ch/mine.alsan/publication
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