Lower Bounds on the Probability of Error for Classical and Classical-Quantum Channels

Dalai, Marco

doi:10.1109/tit.2013.2283794

Cited by 65 publications

(52 citation statements)

References 54 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The error exponent (or reliability function) of c-q channels (see, e.g., Refs. [4][5][6]) corresponds to the case t = 0 where the rate is bounded away from capacity and the error probability vanishes exponentially in n. This is also called the large deviations regime. Moreover, the second-order asymptotics of c-q channels were evaluated by Tomamichel and Tan [7].…”

Section: Introductionmentioning

confidence: 99%

Moderate Deviation Analysis for Classical Communication over Quantum Channels

Chubb

Tan

Tomamichel

2017

Commun. Math. Phys.

View full text Add to dashboard Cite

We analyse families of codes for classical data transmission over quantum channels that have both a vanishing probability of error and a code rate approaching capacity as the code length increases. To characterise the fundamental tradeoff between decoding error, code rate and code length for such codes we introduce a quantum generalisation of the moderate deviation analysis proposed by Altȗg and Wagner as well as Polyanskiy and Verdú. We derive such a tradeoff for classical-quantum (as well as image-additive) channels in terms of the channel capacity and the channel dispersion, giving further evidence that the latter quantity characterises the necessary backoff from capacity when transmitting finite blocks of classical data. To derive these results we also study asymmetric binary quantum hypothesis testing in the moderate deviations regime. Due to the central importance of the latter task, we expect that our techniques will find further applications in the analysis of other quantum information processing tasks.

show abstract

Section: Introductionmentioning

confidence: 99%

Moderate Deviation Analysis for Classical Communication over Quantum Channels

Chubb

Tan

Tomamichel

2017

Commun. Math. Phys.

View full text Add to dashboard Cite

show abstract

“…This particular result was first derived in different forms in the 1960s for classical channels (of different types) and more recently in [11][12][13] for classical-quantum channels. The aim of this paper is to present a detailed and self-contained discussion of the differences between the classical and classical-quantum settings, pointing out connections with an important open problem first suggested by Holevo in [10] and possibly with recent results derived by Mosonyi and Ogawa in [14].…”

Section: Introductionmentioning

confidence: 85%

Some Remarks on Classical and Classical-Quantum Sphere Packing Bounds: Rényi vs. Kullback–Leibler

Dalai

2017

Entropy

Self Cite

View full text Add to dashboard Cite

Abstract:We review the use of binary hypothesis testing for the derivation of the sphere packing bound in channel coding, pointing out a key difference between the classical and the classical-quantum setting. In the first case, two ways of using the binary hypothesis testing are known, which lead to the same bound written in different analytical expressions. The first method historically compares output distributions induced by the codewords with an auxiliary fixed output distribution, and naturally leads to an expression using the Renyi divergence. The second method compares the given channel with an auxiliary one and leads to an expression using the Kullback-Leibler divergence. In the classical-quantum case, due to a fundamental difference in the quantum binary hypothesis testing, these two approaches lead to two different bounds, the first being the "right" one. We discuss the details of this phenomenon, which suggests the question of whether auxiliary channels are used in the optimal way in the second approach and whether recent results on the exact strong-converse exponent in classical-quantum channel coding might play a role in the considered problem.

show abstract

“…Recently Dalai has proven upper bounds on the reliability function of a probabilistic channel that are finite for all rates above at the (logarithmic) Shannon capacity of the underlying confusion graph, in contrast to previous bounds that were finite for rates above log p * (A) [10]. The idea of multiple channels with the same confusion graph plays an important role here.…”

Section: Definition 223mentioning

confidence: 91%

Combinatorial channels from partially ordered sets

Cullina

Kiyavash

2015

2015 IEEE Information Theory Workshop (ITW)

View full text Add to dashboard Cite

A central task of coding theory is the design of schemes to reliably transmit data though space, via communication systems, or through time, via storage systems. Our goal is to identify and exploit structural properties common to a wide variety of coding problems, classical and modern, using the framework of partially ordered sets. We represent adversarial error models as combinatorial channels, form combinatorial channels from posets, identify a structural property of posets that leads to families of channels with the same codes, and bound the size of codes by optimizing over a family of equivalent channels. A large number of previously studied coding problems that fit into this framework. This leads to a new upper bound on the size of s-deletion correcting codes. We use a linear programming framework to obtain spherepacking upper bounds when there is little underlying symmetry in the coding problem. Finally, we introduce and investigate a strong notion of poset homomorphism: locally bijective cover preserving maps. We look for maps of this type to and from the subsequence partial order on q-ary strings.ii Our goal is to identify and exploit structural properties common to a wide variety of coding problems, classical and modern, using the framework of partially ordered sets.In Chapter 2, we introduce the main ideas. We use the notion of a combinatorial channel to represent an adversarial or worst-case error model. We define partially ordered sets (posets), poset homomorphisms, and the other fundamental concepts. We describe a simple method for forming combinatorial channels from posets and identify a structural property of posets that leads to families of channels with the same codes. This leads to ones of our core techniques: bounding the size of codes by optimizing over a family of equivalent channels. In the second half of the chapter, we give examples of previously studied coding problems that fit into this framework. This chapter is based primarily on [1] and also contains some material from [2].In Chapter 3, we apply the methods described in Chapter 2 to the subsequence partial order on q-ary strings. This leads to a new upper bound on the size of s-deletion correcting codes, for fixed s and input length going to infinity. The new bounds improves on the previous best bound whenever the number of deletions is greater than the alphabet size. At the end of this chapter, we look at the partial order on integer compositions. This has many features in common with the subsequence partial order, but many enumerative problems are more tractable. The channels related to the composition partial order also have emerging applications, particularly in DNA storage systems. This chapter is based primarily on [3] and Section 3.8 is based on [1].Chapter 4 covers the linear programming framework for sphere-packing upper bounds. The main contributions are methods for bounding the values 1 of these linear programs when they involve too many variables and constraints to evaluate or even analyze exactly. These methods a...

show abstract

Lower Bounds on the Probability of Error for Classical and Classical-Quantum Channels

Cited by 65 publications

References 54 publications

Moderate Deviation Analysis for Classical Communication over Quantum Channels

Moderate Deviation Analysis for Classical Communication over Quantum Channels

Some Remarks on Classical and Classical-Quantum Sphere Packing Bounds: Rényi vs. Kullback–Leibler

Combinatorial channels from partially ordered sets

Contact Info

Product

Resources

About