Quantum key distribution (QKD) is the first quantum information task to reach the level of mature technology, already fit for commercialization. It aims at the creation of a secret key between authorized partners connected by a quantum channel and a classical authenticated channel. The security of the key can in principle be guaranteed without putting any restriction on an eavesdropper's power. This article provides a concise up-to-date review of QKD, biased toward the practical side. Essential theoretical tools that have been developed to assess the security of the main experimental platforms are presented (discrete-variable, continuous-variable, and distributed-phase-reference protocols).

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The security of practical quantum key distribution

The science of quantum information has arisen over the last two decades centered on the manipulation of individual quanta of information, known as quantum bits or qubits. Quantum computers, quantum cryptography, and quantum teleportation are among the most celebrated ideas that have emerged from this new field. It was realized later on that using continuous-variable quantum information carriers, instead of qubits, constitutes an extremely powerful alternative approach to quantum information processing. This review focuses on continuous-variable quantum information processes that rely on any combination of Gaussian states, Gaussian operations, and Gaussian measurements. Interestingly, such a restriction to the Gaussian realm comes with various benefits, since on the theoretical side, simple analytical tools are available and, on the experimental side, optical components effecting Gaussian processes are readily available in the laboratory. Yet, Gaussian quantum information processing opens the way to a wide variety of tasks and applications, including quantum communication, quantum cryptography, quantum computation, quantum teleportation, and quantum state and channel discrimination. This review reports on the state of the art in this field, ranging from the basic theoretical tools and landmark experimental realizations to the most recent successful developments.

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Gaussian quantum information

Information Theory and Reliable Communication

Random matrix theory has found many applications in physics, statistics and engineering since its inception. Although early developments were motivated by practical experimental problems, random matrices are now used in fields as diverse as Riemann hypothesis, stochastic differential equations, condensed matter physics, statistical physics, chaotic systems, numerical linear algebra, neural networks, multivariate statistics, information theory, signal processing and small-world networks. This article provides a tutorial on random matrices which provides an overview of the theory and brings together in one source the most significant results recently obtained. Furthermore, the application of random matrix theory to the fundamental limits of wireless communication channels is described in depth.

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Random Matrix Theory and Wireless Communications

Recent advancements in iterative processing of channel codes and the development of turbo codes have allowed the communications industry to achieve near-capacity on a single-antenna Gaussian or fading channel with low complexity. We show how these iterative techniques can also be used to achieve near-capacity on a multiple-antenna system where the receiver knows the channel. Combining iterative processing with multiple-antenna channels is particularly challenging because the channel capacities can be a factor of ten or more higher than their single-antenna counterparts. Using a "list" version of the sphere decoder, we provide a simple method to iteratively detect and decode any linear space-time mapping combined with any channel code that can be decoded using so-called "soft" inputs and outputs. We exemplify our technique by directly transmitting symbols that are coded with a channel code; we show that iterative processing with even this simple scheme can achieve near-capacity. We consider both simple convolutional and powerful turbo channel codes and show that excellent performance at very high data rates can be attained with either. We compare our simulation results with Shannon capacity limits for ergodic multiple-antenna channel.

http://www.ece.ualberta.ca/~hcdc/Library/StCommClass/HocTen01.pdf

Achieving near-capacity on a multiple-antenna channel

We present a maximum-likelihood decoding algorithm for an arbitrary lattice code when used over an independent fading channel with perfect channel state information at the receiver. The decoder is based on a bounded distance search among the lattice points falling inside a sphere centered at the received point. By judicious choice of the decoding radius we show that this decoder can be practically used to decode lattice codes of dimension up to 32 in a fading environment.

A universal lattice code decoder for fading channels

The increasing need for high data-rate transmissions over time- or frequency-selective fading channels has drawn attention to modulation schemes with high spectral efficiency such as QAM. With the aim of increasing the "diversity order" of the signal set we consider multidimensional rotated QAM constellations. Very high diversity orders can be achieved and this results in an almost Gaussian performance over the fading channel, This multidimensional modulation scheme is essentially uncoded and enables one to trade diversity for system complexity, at no power or bandwidth expense.

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Signal space diversity: a power- and bandwidth-efficient diversity technique for the Rayleigh fading channel

Recent work on lattices matched to the Rayleigh fading channel has shown how to construct good signal constellations with high spectral efficiency. We present a new family of lattice constellations, based on complex algebraic number fields, which have good performance on Rayleigh fading channels. Some of these lattices also present a reasonable packing density and thus may be used at the same time over a Gaussian channel. Conversely, we show that particular versions of the best lattice packings (D/sub 4/, E/sub 6/, E/sub 8/, K/sub 12/, /spl Lambda//sub 16/, /spl Lambda//sub 24/), constructed from totally complex algebraic cyclotomic fields, present better performance over the Rayleigh fading channel. The practical interest in such signal constellations rises from the need to transmit information at high rates over both terrestrial and satellite links. Some further results in algebraic number theory related to ideals and their factorization are presented and the decoding algorithm used with these lattice constellations are illustrated together with practical results.

Good lattice constellations for both Rayleigh fading and Gaussian channels

The synchronous chip-rate discrete-time CDMA channel with channel coding is analysed and the corresponding factor graph is presented. Iterative joint decoding can be derived in a simple and direct way by applying the sum-product algorithm to the factor graph. Since variables have degree 2, no computation takes place at the variable nodes. Computation at the code nodes is just soft-in soft-out (SISO) decoding, whose output is the extrinsic PMF of the coded symbols. Computation at the channel transition nodes is equivalent to MAP symbol-by-symbol multiuser detection, whose complexity is generally exponential in K. We show that several previously proposed low-complexity algorithms based on interference cancellation (IC) can be derived in a simple direct way by approximating the extrinsic PMF output by the SISO decoders either as a single mass-point PMF (hard decision) or as a Gaussian PDF with the same mean and variance (moment matching). Differently from all previously presented methods (derived from heuristic reasoning), we see clearly that extrinsic rather than a posteriori PMF should be fed back. This yields important advantages in terms of limiting achievable throughput.

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Iterative multiuser joint decoding: unified framework and asymptotic analysis

We build a class of pseudo-random error correcting codes, called generalized low density codes (GLD), from the intersection of two interleaved block codes. GLD code performance approaches the channel capacity limit and the GLD decoder is based on simple and fast SISO (soft input-soft output) decoders of smaller block codes. GLD codes are a special case of Tanner codes and a generalization of Gallager's LDPC codes. It is also proved by an ensemble performance argument that these codes are asymptotically good in the sense of the minimum distance criterion. The flexibility in selecting the parameters of GLD codes makes them suitable for small and large block length forward error correcting schemes.

Joseph J. Boutros

Papers

A universal lattice code decoder for fading channels

Signal space diversity: a power- and bandwidth-efficient diversity technique for the Rayleigh fading channel

Good lattice constellations for both Rayleigh fading and Gaussian channels

Iterative multiuser joint decoding: unified framework and asymptotic analysis

Generalized low density (Tanner) codes