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

Dynamical processes in macroscopic systems are often approximately described by kinetic and hydrodynamic equations. One of the central problems in nonequilibrium statistical mechanics is to understand the approximate validity of these equations starting from a microscopic model. We discuss a variety of classical as well as quantum-mechanical models for which kinetic equations can be derived rigorously. The probabilistic nature of the problem is emphasized: The approximation of the microscopic dynamics by either a kinetic or a hydrodynamic equation can be understood as the approximation of a non-Markovian stochastic process by a Markovian process.

Kinetic equations from Hamiltonian dynamics: Markovian limits

We show how to compute or at least to estimate various capacity-related quantities for bosonic Gaussian channels. Among these are the coherent information, the entanglement-assisted classical capacity, the one-shot classical capacity, and a quantity involving the transpose operation, shown to be a general upper bound on the quantum capacity, even allowing for finite errors. All bounds are explicitly evaluated for the case of a one-mode channel with attenuation or amplification and classical noise.

Evaluating capacities of bosonic Gaussian channels

We show how to compute or at least to estimate various capacity-related quantities for Bosonic Gaussian channels. Among these are the coherent information, the entanglement assisted classical capacity, the one-shot classical capacity, and a new quantity involving the transpose operation, shown to be a general upper bound on the quantum capacity, even allowing for finite errors. All bounds are explicitly evaluated for the case of a one-mode channel with attenuation/amplification and classical noise.

Evaluating capacities of Bosonic Gaussian channels

https://www.math.uci.edu/~brusso/keylphysrep.pdf

Fundamentals of quantum information theory

Given any operator on the testfunction space, the general form of the induced completely positive map of the C*-algebra of the canonical commutation relations is characterized.

Completely positive maps on the CCR-algebra

Completely positive quasi-free maps of the CCR-algebra

A new characterization of equilibrium states for classical lattice systems is given in terms of correlation inequalities. Their physical meaning is found to express thermodynamic stability. We demonstrate the applicability of the inequalities in specific models.

Energy-Entropy Inequalities for Classical Lattice Systems

For quantum systems we develop a new method, based on a general energy‐entropy inequality, to rule out spontaneous breaking of symmetries. The main advantage of our scheme consists in its clear‐cut physical significance and its new areas of applicability; in particular we can handle discrete symmetry groups as well as continuous ones. Finally a few illustrations are discussed.

Quantum energy-entropy inequalities: a new method for proving the absence of symmetry breaking

For quantum systems as well as for classical continuous systems energetic stability is defined. It is proved that stability, supplemented with a cluster property, characterizes equilibrium states.

Paul Vanheuverzwijn

Papers

Completely positive maps on the CCR-algebra

Completely positive quasi-free maps of the CCR-algebra

Energy-Entropy Inequalities for Classical Lattice Systems

Quantum energy-entropy inequalities: a new method for proving the absence of symmetry breaking

Energetically stable systems