Bounds on Information Combining With Quantum Side Information

"Bounds on information combining" are entropic inequalities that determine how the information (entropy) of a set of random variables can change when these are combined in certain prescribed ways. Such bounds play an important role in classical information theory, particularly in coding and Shannon theory; entropy power inequalities are special instances of them. The arguably most elementary kind of information combining is the addition of two binary random variables (a CNOT gate), and the resulting quantities play an important role in Belief propagation and Polar coding. We investigate this problem in the setting where quantum side information is available, which has been recognized as a hard setting for entropy power inequalities. Our main technical result is a non-trivial, and close to optimal, lower bound on the combined entropy, which can be seen as an almost optimal "quantum Mrs. Gerber's Lemma". Our proof uses three main ingredients: (1) a new bound on the concavity of von Neumann entropy, which is tight in the regime of low pairwise state fidelities; (2) the quantitative improvement of strong subadditivity due to Fawzi-Renner, in which we manage to handle the minimization over recovery maps; (3) recent duality results on classical-quantum-channels due to Renes et al. We furthermore present conjectures on the optimal lower and upper bounds under quantum side information, supported by interesting analytical observations and strong numerical evidence. We finally apply our bounds to Polar coding for binary-input classical-quantum channels, and show the following three results: (A) Even non-stationary channels polarize under the polar transform. (B) The blocklength required to approach the symmetric capacity scales at most sub-exponentially in the gap to capacity. (C) Under the aforementioned lower bound conjecture, a blocklength polynomial in the gap suffices.Comment: 23 pages, 6 figures; v2: small correction

Source Polarization

The notion of source polarization is introduced and investigated. This complements the earlier work on channel polarization. An application to Slepian-Wolf coding is also considered. The paper is restricted to the case of binary alphabets. Extension of results to non-binary alphabets is discussed briefly.Comment: To be presented at the IEEE 2010 International Symposium on Information Theory

General Strong Polarization

Arikan's exciting discovery of polar codes has provided an altogether new way to efficiently achieve Shannon capacity. Given a (constant-sized) invertible matrix $M$, a family of polar codes can be associated with this matrix and its ability to approach capacity follows from the {\em polarization} of an associated $[0,1]$-bounded martingale, namely its convergence in the limit to either $0$ or $1$. Arikan showed polarization of the martingale associated with the matrix $G_2 = \left(\begin{matrix} 1& 0 1& 1\end{matrix}\right)$ to get capacity achieving codes. His analysis was later extended to all matrices $M$ that satisfy an obvious necessary condition for polarization. While Arikan's theorem does not guarantee that the codes achieve capacity at small blocklengths, it turns out that a "strong" analysis of the polarization of the underlying martingale would lead to such constructions. Indeed for the martingale associated with $G_2$ such a strong polarization was shown in two independent works ([Guruswami and Xia, IEEE IT '15] and [Hassani et al., IEEE IT '14]), resolving a major theoretical challenge of the efficient attainment of Shannon capacity. In this work we extend the result above to cover martingales associated with all matrices that satisfy the necessary condition for (weak) polarization. In addition to being vastly more general, our proofs of strong polarization are also simpler and modular. Specifically, our result shows strong polarization over all prime fields and leads to efficient capacity-achieving codes for arbitrary symmetric memoryless channels. We show how to use our analyses to achieve exponentially small error probabilities at lengths inverse polynomial in the gap to capacity. Indeed we show that we can essentially match any error probability with lengths that are only inverse polynomial in the gap to capacity.Comment: 73 pages, 2 figures. The final version appeared in JACM. This paper combines results presented in preliminary form at STOC 2018 and RANDOM 201

Universal Polarization

A method to polarize channels universally is introduced. The method is based on combining two distinct channels in each polarization step, as opposed to Arikan's original method of combining identical channels. This creates an equal number of only two types of channels, one of which becomes progressively better as the other becomes worse. The locations of the good polarized channels are independent of the underlying channel, guaranteeing universality. Polarizing the good channels further with Arikan's method results in universal polar codes of rate 1/2. The method is generalized to construct codes of arbitrary rates. It is also shown that the less noisy ordering of channels is preserved under polarization, and thus a good polar code for a given channel will perform well over a less noisy one.Comment: Submitted to the IEEE Transactions on Information Theor

Countably Infinite Multilevel Source Polarization for Non-Stationary Erasure Distributions

Polar transforms are central operations in the study of polar codes. This paper examines polar transforms for non-stationary memoryless sources on possibly infinite source alphabets. This is the first attempt of source polarization analysis over infinite alphabets. The source alphabet is defined to be a Polish group, and we handle the Ar{\i}kan-style two-by-two polar transform based on the group. Defining erasure distributions based on the normal subgroup structure, we give recursive formulas of the polar transform for our proposed erasure distributions. As a result, the recursive formulas lead to concrete examples of multilevel source polarization with countably infinite levels when the group is locally cyclic. We derive this result via elementary techniques in lattice theory.Comment: 12 pages, 1 figure, a short version has been accepted by the 2019 IEEE International Symposium on Information Theory (ISIT2019

Source and Channel Polarization over Finite Fields and Reed-Solomon Matrices

Polarization phenomenon over any finite field $\mathbb{F}_{q}$ with size $q$ being a power of a prime is considered. This problem is a generalization of the original proposal of channel polarization by Arikan for the binary field, as well as its extension to a prime field by Sasoglu, Telatar, and Arikan. In this paper, a necessary and sufficient condition of a matrix over a finite field $\mathbb{F}_q$ is shown under which any source and channel are polarized. Furthermore, the result of the speed of polarization for the binary alphabet obtained by Arikan and Telatar is generalized to arbitrary finite field. It is also shown that the asymptotic error probability of polar codes is improved by using the Reed-Solomon matrix, which can be regarded as a natural generalization of the $2\times 2$ binary matrix used in the original proposal by Arikan.Comment: 17 pages, 3 figures, accepted for publication in the IEEE Transactions on Information Theor

R\'enyi Bounds on Information Combining

Bounds on information combining are entropic inequalities that determine how the information, or entropy, of a set of random variables can change when they are combined in certain prescribed ways. Such bounds play an important role in information theory, particularly in coding and Shannon theory. The arguably most elementary kind of information combining is the addition of two binary random variables, i.e. a CNOT gate, and the resulting quantities are fundamental when investigating belief propagation and polar coding. In this work we will generalize the concept to R\'enyi entropies. We give optimal bounds on the conditional R\'enyi entropy after combination, based on a certain convexity or concavity property and discuss when this property indeed holds. Since there is no generally agreed upon definition of the conditional R\'enyi entropy, we consider four different versions from the literature. Finally, we discuss the application of these bounds to the polarization of R\'enyi entropies under polar codes.Comment: 14 pages, accepted for presentation at ISIT 202