Strong Converse and Stein's Lemma in the Quantum Hypothesis Testing

The hypothesis testing problem of two quantum states is treated. We show a new inequality between the error of the first kind and the second kind, which complements the result of Hiai and Petz to establish the quantum version of Stein's lemma. The inequality is also used to show a bound on the first kind error when the power exponent for the second kind error exceeds the quantum relative entropy, and the bound yields the strong converse in the quantum hypothesis testing. Finally, we discuss the relation between the bound and the power exponent derived by Han and Kobayashi in the classical hypothesis testing.Comment: LaTeX, 12 pages, submitted to IEEE Trans. Inform. Theor

The AWGN Red Alert Problem

Consider the following unequal error protection scenario. One special message, dubbed the "red alert" message, is required to have an extremely small probability of missed detection. The remainder of the messages must keep their average probability of error and probability of false alarm below a certain threshold. The goal then is to design a codebook that maximizes the error exponent of the red alert message while ensuring that the average probability of error and probability of false alarm go to zero as the blocklength goes to infinity. This red alert exponent has previously been characterized for discrete memoryless channels. This paper completely characterizes the optimal red alert exponent for additive white Gaussian noise channels with block power constraints.Comment: 13 pages, 10 figures, To appear in IEEE Transactions on Information Theor

Efficient sphere-covering and converse measure concentration via generalized coding theorems

Suppose A is a finite set equipped with a probability measure P and let M be a ``mass'' function on A. We give a probabilistic characterization of the most efficient way in which A^n can be almost-covered using spheres of a fixed radius. An almost-covering is a subset C_n of A^n, such that the union of the spheres centered at the points of C_n has probability close to one with respect to the product measure P^n. An efficient covering is one with small mass M^n(C_n); n is typically large. With different choices for M and the geometry on A our results give various corollaries as special cases, including Shannon's data compression theorem, a version of Stein's lemma (in hypothesis testing), and a new converse to some measure concentration inequalities on discrete spaces. Under mild conditions, we generalize our results to abstract spaces and non-product measures.Comment: 29 pages. See also http://www.stat.purdue.edu/~yiannis

On Identifying a Massive Number of Distributions

Finding the underlying probability distributions of a set of observed sequences under the constraint that each sequence is generated i.i.d by a distinct distribution is considered. The number of distributions, and hence the number of observed sequences, are let to grow with the observation blocklength $n$. Asymptotically matching upper and lower bounds on the probability of error are derived.Comment: Under Submissio

A Rate-Distortion Exponent Approach to Multiple Decoding Attempts for Reed-Solomon Codes

Algorithms based on multiple decoding attempts of Reed-Solomon (RS) codes have recently attracted new attention. Choosing decoding candidates based on rate-distortion (R-D) theory, as proposed previously by the authors, currently provides the best performance-versus-complexity trade-off. In this paper, an analysis based on the rate-distortion exponent (RDE) is used to directly minimize the exponential decay rate of the error probability. This enables rigorous bounds on the error probability for finite-length RS codes and leads to modest performance gains. As a byproduct, a numerical method is derived that computes the rate-distortion exponent for independent non-identical sources. Analytical results are given for errors/erasures decoding.Comment: accepted for presentation at 2010 IEEE International Symposium on Information Theory (ISIT 2010), Austin TX, US