380 research outputs found
On the calculation of the minimax-converse of the channel coding problem
A minimax-converse has been suggested for the general channel coding problem
by Polyanskiy etal. This converse comes in two flavors. The first flavor is
generally used for the analysis of the coding problem with non-vanishing error
probability and provides an upper bound on the rate given the error
probability. The second flavor fixes the rate and provides a lower bound on the
error probability. Both converses are given as a min-max optimization problem
of an appropriate binary hypothesis testing problem. The properties of the
first converse were studies by Polyanskiy and a saddle point was proved. In
this paper we study the properties of the second form and prove that it also
admits a saddle point. Moreover, an algorithm for the computation of the saddle
point, and hence the bound, is developed. In the DMC case, the algorithm runs
in a polynomial time.Comment: Extended version of a submission to ISIT 201
A Minimax Converse for Quantum Channel Coding
We prove a one-shot "minimax" converse bound for quantum channel coding
assisted by positive partial transpose channels between sender and receiver.
The bound is similar in spirit to the converse by Polyanskiy, Poor, and Verdu
[IEEE Trans. Info. Theory 56, 2307-2359 (2010)] for classical channel coding,
and also enjoys the saddle point property enabling the order of optimizations
to be interchanged. Equivalently, the bound can be formulated as a semidefinite
program satisfying strong duality. The convex nature of the bound implies
channel symmetries can substantially simplify the optimization, enabling us to
explicitly compute the finite blocklength behavior for several simple qubit
channels. In particular, we find that finite blocklength converse statements
for the classical erasure channel apply to the assisted quantum erasure
channel, while bounds for the classical binary symmetric channel apply to both
the assisted dephasing and depolarizing channels. This implies that these qubit
channels inherit statements regarding the asymptotic limit of large
blocklength, such as the strong converse or second-order converse rates, from
their classical counterparts. Moreover, for the dephasing channel, the finite
blocklength bounds are as tight as those for the classical binary symmetric
channel, since coding for classical phase errors yields equivalently-performing
unassisted quantum codes.Comment: merged with arXiv:1504.04617 version 1 ; see version
The third-order term in the normal approximation for singular channels
For a singular and symmetric discrete memoryless channel with positive
dispersion, the third-order term in the normal approximation is shown to be
upper bounded by a constant. This finding completes the characterization of the
third-order term for symmetric discrete memoryless channels. The proof method
is extended to asymmetric and singular channels with constant composition
codes, and its connection to existing results, as well as its limitation in the
error exponents regime, are discussed.Comment: Submitted to IEEE Trans. Inform. Theor
Multiplicativity of completely bounded -norms implies a strong converse for entanglement-assisted capacity
The fully quantum reverse Shannon theorem establishes the optimal rate of
noiseless classical communication required for simulating the action of many
instances of a noisy quantum channel on an arbitrary input state, while also
allowing for an arbitrary amount of shared entanglement of an arbitrary form.
Turning this theorem around establishes a strong converse for the
entanglement-assisted classical capacity of any quantum channel. This paper
proves the strong converse for entanglement-assisted capacity by a completely
different approach and identifies a bound on the strong converse exponent for
this task. Namely, we exploit the recent entanglement-assisted "meta-converse"
theorem of Matthews and Wehner, several properties of the recently established
sandwiched Renyi relative entropy (also referred to as the quantum Renyi
divergence), and the multiplicativity of completely bounded -norms due to
Devetak et al. The proof here demonstrates the extent to which the Arimoto
approach can be helpful in proving strong converse theorems, it provides an
operational relevance for the multiplicativity result of Devetak et al., and it
adds to the growing body of evidence that the sandwiched Renyi relative entropy
is the correct quantum generalization of the classical concept for all
.Comment: 21 pages, final version accepted for publication in Communications in
Mathematical Physic
Random Access Channel Coding in the Finite Blocklength Regime
Consider a random access communication scenario over a channel whose
operation is defined for any number of possible transmitters. Inspired by the
model recently introduced by Polyanskiy for the Multiple Access Channel (MAC)
with a fixed, known number of transmitters, we assume that the channel is
invariant to permutations on its inputs, and that all active transmitters
employ identical encoders. Unlike Polyanskiy, we consider a scenario where
neither the transmitters nor the receiver know which transmitters are active.
We refer to this agnostic communication setup as the Random Access Channel, or
RAC. Scheduled feedback of a finite number of bits is used to synchronize the
transmitters. The decoder is tasked with determining from the channel output
the number of active transmitters () and their messages but not which
transmitter sent which message. The decoding procedure occurs at a time
depending on the decoder's estimate of the number of active transmitters,
, thereby achieving a rate that varies with the number of active
transmitters. Single-bit feedback at each time , enables all
transmitters to determine the end of one coding epoch and the start of the
next. The central result of this work demonstrates the achievability on a RAC
of performance that is first-order optimal for the MAC in operation during each
coding epoch. While prior multiple access schemes for a fixed number of
transmitters require simultaneous threshold rules, the proposed
scheme uses a single threshold rule and achieves the same dispersion.Comment: Presented at ISIT18', submitted to IEEE Transactions on Information
Theor
Strong converse exponents for the feedback-assisted classical capacity of entanglement-breaking channels
Quantum entanglement can be used in a communication scheme to establish a
correlation between successive channel inputs that is impossible by classical
means. It is known that the classical capacity of quantum channels can be
enhanced by such entangled encoding schemes, but this is not always the case.
In this paper, we prove that a strong converse theorem holds for the classical
capacity of an entanglement-breaking channel even when it is assisted by a
classical feedback link from the receiver to the transmitter. In doing so, we
identify a bound on the strong converse exponent, which determines the
exponentially decaying rate at which the success probability tends to zero, for
a sequence of codes with communication rate exceeding capacity. Proving a
strong converse, along with an achievability theorem, shows that the classical
capacity is a sharp boundary between reliable and unreliable communication
regimes. One of the main tools in our proof is the sandwiched Renyi relative
entropy. The same method of proof is used to derive an exponential bound on the
success probability when communicating over an arbitrary quantum channel
assisted by classical feedback, provided that the transmitter does not use
entangled encoding schemes.Comment: 24 pages, 2 figures, v4: final version accepted for publication in
Problems of Information Transmissio
Saddle Point in the Minimax Converse for Channel Coding
A minimax metaconverse has recently been proposed as a simultaneous generalization of a number of classical results and a tool for the nonasymptotic analysis. In this paper, it is shown that the order of optimizing the input and output distributions can be interchanged without affecting the bound. In the course of the proof, a number of auxiliary results of separate interest are obtained. In particular, it is shown that the optimization problem is convex and can be solved in many cases by the symmetry considerations. As a consequence, it is demonstrated that in the latter cases, the (multiletter) input distribution in information-spectrum (Verdú-Han) converse bound can be taken to be a (memoryless) product of single-letter ones. A tight converse for the binary erasure channel is rederived by computing the optimal (nonproduct) output distribution. For discrete memoryless channels, a conjecture of Poor and Verdú regarding the tightness of the information spectrum bound on the error exponents is resolved in the negative. Concept of the channel symmetry group is established and relations with the definitions of symmetry by Gallager and Dobrushin are investigated.National Science Foundation (U.S.) (Center for Science of Information, under Grant CCF-0939370
On converse bounds for classical communication over quantum channels
We explore several new converse bounds for classical communication over
quantum channels in both the one-shot and asymptotic regimes. First, we show
that the Matthews-Wehner meta-converse bound for entanglement-assisted
classical communication can be achieved by activated, no-signalling assisted
codes, suitably generalizing a result for classical channels. Second, we derive
a new efficiently computable meta-converse on the amount of classical
information unassisted codes can transmit over a single use of a quantum
channel. As applications, we provide a finite resource analysis of classical
communication over quantum erasure channels, including the second-order and
moderate deviation asymptotics. Third, we explore the asymptotic analogue of
our new meta-converse, the -information of the channel. We show that
its regularization is an upper bound on the classical capacity, which is
generally tighter than the entanglement-assisted capacity and other known
efficiently computable strong converse bounds. For covariant channels we show
that the -information is a strong converse bound.Comment: v3: published version; v2: 18 pages, presentation and results
improve
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
Finite Blocklength Analysis of Gaussian Random Coding in AWGN Channels under Covert Constraint
This paper considers the achievability and converse bounds on the maximal
channel coding rate at a given blocklength and error probability over AWGN
channels. The problem stems from covert communication with Gaussian codewords.
By re-visiting [18], we first present new and more general achievability bounds
for random coding schemes under maximal or average probability of error
requirements. Such general bounds are then applied to covert communication in
AWGN channels where codewords are generated from Gaussian distribution while
meeting the maximal power constraint. Further comparison is made between the
new achievability bounds and existing one with deterministic codebooks.Comment: 18 page
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