We report on the implementation of a reverse-reconciliated coherent-state continuous-variable quantum key distribution system, with which we generated secret keys at a rate of more than 2 kb/s over 25 km of optical fiber. Time multiplexing is used to transmit both the signal and phase reference in the same optical fiber. Our system includes all experimental aspects required for a field implementation of a quantum key distribution setup. Real-time reverse reconciliation is achieved by using fast and efficient LDPC error correcting codes.

We report on both theoretical and experimental aspects of a fully implemented quantum key distribution device with coherent states. This system features a final key rate of more than 2 kb/s over 25 km of optical fiber. It comprises all required elements for field operation: a compact optical setup, a fast secret bit extraction using efficient LDPC codes, privacy amplification algorithms and a classical channel software. Both hardware and software are operated in real time.

In a series of interesting papers G. C. Hegerfeldt has shown that quantum systems with positive energy initially localized in a finite region, immediately develop infinite tails. In our paper Hegerfeldt's theorem is analysed using quantum and classical wave packets. We show that Hegerfeldt's conclusion remains valid in classical physics. No violation of Einstein's causality is ever involved. Using only positive frequencies, complex wave packets are constructed which at t = 0 are real and finitely localized and which, furthemore, are superpositions of two nonlocal wave packets. The nonlocality is initially cancelled by destructive interference. However this cancellation becomes incomplete at arbitrary times immediately afterwards. In agreement with relativity the two nonlocal wave packets move with the velocity of light, in opposite directions.

We address the classical capacity of a quantum bosonic memory channel with additive noise, subject to an input energy constraint. The memory is modeled by correlated noise emerging from a Gauss-Markov process. Under reasonable assumptions, we show that the optimal modulation results from a "quantum water-filling" solution above a certain input energy threshold, similar to the optimal modulation for parallel classical Gaussian channels. We also derive analytically the optimal multimode input state above this threshold, which enables us to compute the capacity of this memory channel in the limit of an infinite number of modes. The method can also be applied to a more general noise environment which is constructed by a stationary Gauss process. The extension of our results to the case of broadband bosonic channels with colored Gaussian noise should also be straightforward.

According to Hudson's theorem, any pure quantum state with a positive Wigner function is necessarily a Gaussian state. Here, we make a step toward the extension of this theorem to mixed quantum states by finding upper and lower bounds on the degree of non-Gaussianity of states with positive Wigner functions. The bounds are expressed in the form of parametric functions relating the degree of non-Gaussianity of a state, its purity, and the purity of the Gaussian state characterized by the same covariance matrix. Although our bounds are not tight, they permit us to visualize the set of states with positive Wigner functions. The Wigner representation of quantum states ͓1͔, which is realized by joint quasiprobability distributions of canonically conjugate variables in phase space, has a specific property which differentiates it from a true probability distribution: it can attain negative values. Among pure states, it was proven by Hudson ͓2͔ ͑and later generalized to multimode quantum systems by Soto and Claverie ͓3͔͒ that the only states which have non-negative Wigner functions are Gaussian states ͓4͔. The question that naturally arises ͓2͔ is whether this theorem can be extended to mixed states, among which not only Gaussian states may possess a positive Wigner function. A logical extension of the theorem would be a complete characterization of the convex set of states with positive Wigner function. Although this question can be approached by using the notion of Wigner spectrum ͓5͔, a simple and operational extension of Hudson's theorem has not yet been achieved due to the mathematical complications which emerge when dealing with states with positive Wigner functions ͓5͔.Motivated by the increasing interest for non-Gaussian states in continuous-variable quantum information theory ͑see, e.g., ͓6͔͒ and the need for a better understanding of the de-Gaussification procedures for mixed states ͑see, e.g., ͓7͔͒, we attempt here an exploration of the set of states with positive Wigner functions using Gaussian states as a reference. More precisely, we consider the subset of such states that have the same covariance matrix as a reference Gaussian state. We obtain a partial solution to the problem by analytically deriving necessary conditions ͑bounds͒ on a measure of non-Gaussianity for a state to have a positive Wigner function. This set of conditions bounds a region in a threedimensional space with coordinates being the purity of the state, the purity of the corresponding Gaussian state, and the non-Gaussianity. As intuitively expected, the maximum degree of non-Gaussianity increases with a decrease in the purity of both the state and its Gaussian corresponding state.Before deriving the main results of this paper, let us recall a convenient representation of the trace of the product of two one-mode quantum states, and Ј, in terms of the Wigner representation ͓8͔,where W is the Wigner function of the state . For example, the purity of a state, ͓͔ =Tr͑ 2 ͒, may be calculated with the help of this formula. For a state with a G...