63 research outputs found
Quantum Sphere-Packing Bounds with Polynomial Prefactors
Β© 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
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
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
The Sphere Packing Bound For Memoryless Channels
Sphere packing bounds (SPBs) ---with prefactors that are polynomial in the
block length--- are derived for codes on two families of memoryless channels
using Augustin's method: (possibly non-stationary) memoryless channels with
(possibly multiple) additive cost constraints and stationary memoryless
channels with convex constraints on the composition (i.e. empirical
distribution, type) of the input codewords. A variant of Gallager's bound is
derived in order to show that these sphere packing bounds are tight in terms of
the exponential decay rate of the error probability with the block length under
mild hypotheses.Comment: 29 page
The Third-Order Term in the Normal Approximation for the AWGN Channel
This paper shows that, under the average error probability formalism, the
third-order term in the normal approximation for the additive white Gaussian
noise channel with a maximal or equal power constraint is at least . This matches the upper bound derived by
Polyanskiy-Poor-Verd\'{u} (2010).Comment: 13 pages, 1 figur
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
Density of Spherically-Embedded Stiefel and Grassmann Codes
The density of a code is the fraction of the coding space covered by packing
balls centered around the codewords. This paper investigates the density of
codes in the complex Stiefel and Grassmann manifolds equipped with the chordal
distance. The choice of distance enables the treatment of the manifolds as
subspaces of Euclidean hyperspheres. In this geometry, the densest packings are
not necessarily equivalent to maximum-minimum-distance codes. Computing a
code's density follows from computing: i) the normalized volume of a metric
ball and ii) the kissing radius, the radius of the largest balls one can pack
around the codewords without overlapping. First, the normalized volume of a
metric ball is evaluated by asymptotic approximations. The volume of a small
ball can be well-approximated by the volume of a locally-equivalent tangential
ball. In order to properly normalize this approximation, the precise volumes of
the manifolds induced by their spherical embedding are computed. For larger
balls, a hyperspherical cap approximation is used, which is justified by a
volume comparison theorem showing that the normalized volume of a ball in the
Stiefel or Grassmann manifold is asymptotically equal to the normalized volume
of a ball in its embedding sphere as the dimension grows to infinity. Then,
bounds on the kissing radius are derived alongside corresponding bounds on the
density. Unlike spherical codes or codes in flat spaces, the kissing radius of
Grassmann or Stiefel codes cannot be exactly determined from its minimum
distance. It is nonetheless possible to derive bounds on density as functions
of the minimum distance. Stiefel and Grassmann codes have larger density than
their image spherical codes when dimensions tend to infinity. Finally, the
bounds on density lead to refinements of the standard Hamming bounds for
Stiefel and Grassmann codes.Comment: Two-column version (24 pages, 6 figures, 4 tables). To appear in IEEE
Transactions on Information Theor
The Sphere Packing Bound via Augustin's Method
A sphere packing bound (SPB) with a prefactor that is polynomial in the block
length is established for codes on a length product channel
assuming that the maximum order Renyi capacity among the component
channels, i.e. , is . The
reliability function of the discrete stationary product channels with feedback
is bounded from above by the sphere packing exponent. Both results are proved
by first establishing a non-asymptotic SPB. The latter result continues to hold
under a milder stationarity hypothesis.Comment: 30 pages. An error in the statement of Lemma 2 is corrected. The
change is inconsequential for the rest of the pape
Moderate deviation analysis for classical-quantum channels and quantum hypothesis testing
Β© 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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