Polar Codes for Arbitrary Classical-Quantum Channels and Arbitrary cq-MACs

We prove polarization theorems for arbitrary classical-quantum (cq) channels. The input alphabet is endowed with an arbitrary Abelian group operation and an Ar{\i}kan-style transformation is applied using this operation. It is shown that as the number of polarization steps becomes large, the synthetic cq-channels polarize to deterministic homomorphism channels which project their input to a quotient group of the input alphabet. This result is used to construct polar codes for arbitrary cq-channels and arbitrary classical-quantum multiple access channels (cq-MAC). The encoder can be implemented in $O(N\log N)$ operations, where $N$ is the blocklength of the code. A quantum successive cancellation decoder for the constructed codes is proposed. It is shown that the probability of error of this decoder decays faster than $2^{-N^{\beta}}$ for any $\beta<\frac{1}{2}$.Comment: 30 pages. Submitted to IEEE Trans. Inform. Theory and in part to ISIT201

Polarization and Channel Ordering: Characterizations and Topological Structures

Information theory is the field in which we study the fundamental limitations of communication. Shannon proved in 1948 that there exists a maximum rate, called capacity, at which we can reliably communicate information through a given channel. However, Shannon did not provide an explicit construction of a practical coding scheme that achieves the capacity. Polar coding, invented by Arikan, is the first low-complexity coding technique that achieves the capacity of binary-input memoryless symmetric channels. The construction of these codes is based on a phenomenon called polarization. The study of polar codes and their generalization to arbitrary channels is the subject of polarization theory, a subfield of information and coding theories. This thesis consists of two parts. In the first part, we provide solutions to several open problems in polarization theory. The first open problem that we consider is to determine the binary operations that always lead to polarization when they are used in Arikan-style constructions. In order to solve this problem, we develop an ergodic theory for binary operations. This theory is used to provide a necessary and sufficient condition that characterizes the polarizing binary operations, both in the single-user and the multiple-access settings. We prove that the exponent of a polarizing binary operation cannot exceed 1/2. Furthermore, we show that the exponent of an arbitrary quasigroup operation is exactly 1/2. This implies that quasigroup operations are among the best polarizing binary operations. One drawback of polarization in the multiple-access setting is that it sometimes induces a loss in the symmetric capacity region of a given multiple-access channel (MAC). An open problem in MAC polarization theory is to determine all the MACs that do not lose any part of their capacity region by polarization. Using Fourier analysis, we solve this problem by providing a single-letter necessary and sufficient condition that characterizes all these MACs in the general setting where we have an arbitrary number of users, and each user uses an arbitrary Abelian group operation on his input alphabet. We also study the polarization of classical-quantum (cq) channels. The input alphabet is endowed with an arbitrary Abelian group operation, and an Arikan-style transformation is applied using this operation. We show that as the number of polarization steps becomes large, the synthetic cq-channels polarize to deterministic homomorphism channels that project their input to a quotient group of the input alphabet. This result is used to construct polar codes for arbitrary cq-channels and arbitrary classical-quantum multiple-access channels (cq-MAC). In the second part of this thesis, we investigate several problems that are related to three orderings of communication channels: degradedness, input-degradedness, and the Shannon ordering. We provide several characterizations for the input-degradedness and the Shannon ordering. Two channels are said to be equivalent if they are degraded from each other. Input-equivalence and Shannon-equivalence between channels are similarly defined. We construct and study several topologies on the quotients of the spaces of discrete memoryless channels (DMC) by the equivalence, the input-equivalence and the Shannon-equivalence relations. Finally, we prove the continuity of several channel parameters and operations under various DMC topologies

Polar codes in network quantum information theory

Polar coding is a method for communication over noisy classical channels which is provably capacity-achieving and has an efficient encoding and decoding. Recently, this method has been generalized to the realm of quantum information processing, for tasks such as classical communication, private classical communication, and quantum communication. In the present work, we apply the polar coding method to network quantum information theory, by making use of recent advances for related classical tasks. In particular, we consider problems such as the compound multiple access channel and the quantum interference channel. The main result of our work is that it is possible to achieve the best known inner bounds on the achievable rate regions for these tasks, without requiring a so-called quantum simultaneous decoder. Thus, our work paves the way for developing network quantum information theory further without requiring a quantum simultaneous decoder.Comment: 18 pages, 2 figures, v2: 10 pages, double column, version accepted for publicatio

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

Polar Codes for CQ Channels: Decoding via Belief-Propagation with Quantum Messages

This paper considers the design and decoding of polar codes for general classical-quantum (CQ) channels. It focuses on decoding via belief-propagation with quantum messages (BPQM) and, in particular, the idea of paired-measurement BPQM (PM-BPQM) decoding. Since the PM-BPQM decoder admits a classical density evolution (DE) analysis, one can use DE to design a polar code for any CQ channel and then efficiently compute the trade-off between code rate and error probability. We have also implemented and tested a classical simulation of our PM-BPQM decoder for polar codes. While the decoder can be implemented efficiently on a quantum computer, simulating the decoder on a classical computer actually has exponential complexity. Thus, simulation results for the decoder are somewhat limited and are included primarily to validate our theoretical results

Polar Codes for Quantum Reading

Quantum readout provides a general framework for formulating statistical discrimination of quantum channels. Several paths have been taken for such this problem. However, there is much to be done in the avenue of optimizing channel discrimination using classical codes. At least two open questions can be pointed out: how to construct low complexity encoding schemes that are interesting for channel discrimination and, more importantly, how to develop capacity-achieving protocols. This paper aims at presenting a solution to these questions using polar codes. Firstly, we characterize the information rate and reliability parameter of the channels under polar encoding. We also show that the error probability of the scheme proposed decays exponentially with the square root of the code length. Secondly, an analysis of the optimal quantum states to be used as probes is given