Reconciliation of a Quantum-Distributed Gaussian Key

Two parties, Alice and Bob, wish to distill a binary secret key out of a list of correlated variables that they share after running a quantum key distribution protocol based on continuous-spectrum quantum carriers. We present a novel construction that allows the legitimate parties to get equal bit strings out of correlated variables by using a classical channel, with as few leaked information as possible. This opens the way to securely correcting non-binary key elements. In particular, the construction is refined to the case of Gaussian variables as it applies directly to recent continuous-variable protocols for quantum key distribution.Comment: 8 pages, 4 figures. Submitted to the IEEE for possible publication. Revised version to improve its clarit

Cloning and Cryptography with Quantum Continuous Variables

The cloning of quantum variables with continuous spectra is investigated. We define a Gaussian 1-to-2 cloning machine, which copies equally well two conjugate variables such as position and momentum or the two quadrature components of a light mode. The resulting cloning fidelity for coherent states, namely $F=2/3$, is shown to be optimal. An asymmetric version of this Gaussian cloner is then used to assess the security of a continuous-variable quantum key distribution scheme that allows two remote parties to share a Gaussian key. The information versus disturbance tradeoff underlying this continuous quantum cryptographic scheme is then analyzed for the optimal individual attack. Methods to convert the resulting Gaussian keys into secret key bits are also studied. The extension of the Gaussian cloner to optimal $N$-to-$M$ continuous cloners is then discussed, and it is shown how to implement these cloners for light modes, using a phase-insensitive optical amplifier and beam splitters. Finally, a phase-conjugated inputs $(N,N')$-to-$(M,M')$ continuous cloner is defined, yielding $M$ clones and $M'$ anticlones from $N$ replicas of a coherent state and $N'$ replicas of its phase-conjugate (with $M'-M=N'-N$). This novel kind of cloners is shown to outperform the standard $N$-to-$M$ cloners in some situations.Comment: 8 pages, 3 figures, submitted to the special issue of the European Physical Journal D on "Quantum interference and cryptographic keys: novel physics and advancing technologies", proceedings of the conference QUICK 2001, Corsica, April 7-13 2001. Minor correction, references adde

Quadratic Hermite-Pade approximation to the exponential function: a Riemann-Hilbert approach

We investigate the asymptotic behavior of the polynomials p, q, r of degrees n in type I Hermite-Pade approximation to the exponential function, defined by p(z)e^{-z}+q(z)+r(z)e^{z} = O(z^{3n+2}) as z -> 0. These polynomials are characterized by a Riemann-Hilbert problem for a 3x3 matrix valued function. We use the Deift-Zhou steepest descent method for Riemann-Hilbert problems to obtain strong uniform asymptotics for the scaled polynomials p(3nz), q(3nz), and r(3nz) in every domain in the complex plane. An important role is played by a three-sheeted Riemann surface and certain measures and functions derived from it. Our work complements recent results of Herbert Stahl.Comment: 60 pages, 13 figure

Side-Information Coding with Turbo Codes and its Application to Quantum Key Distribution

Turbo coding is a powerful class of forward error correcting codes, which can achieve performances close to the Shannon limit. The turbo principle can be applied to the problem of side-information source coding, and we investigate here its application to the reconciliation problem occurring in a continuous-variable quantum key distribution protocol.Comment: 3 pages, submitted to ISITA 200

Asymptotics for Hermite-Pade rational approximants for two analytic functions with separated pairs of branch points (case of genus 0)

We investigate the asymptotic behavior for type II Hermite-Pade approximation to two functions, where each function has two branch points and the pairs of branch points are separated. We give a classification of the cases such that the limiting counting measures for the poles of the Hermite-Pade approximants are described by an algebraic function of order 3 and genus 0. This situation gives rise to a vector-potential equilibrium problem for three measures and the poles of the common denominator are asymptotically distributed like one of these measures. We also work out the strong asymptotics for the corresponding Hermite-Pade approximants by using a 3x3 Riemann-Hilbert problem that characterizes this Hermite-Pade approximation problem.Comment: 102 pages, 31 figure