Quantum cryptography could well be the first application of quantum mechanics at the individual quanta level. The very fast progress in both theory and experiments over the recent years are reviewed, with emphasis on open questions and technological issues.

Quantum Cryptography

Preface. About the Editors. Part I: Quantum Computing. 1. Quantum computing with qubits S.L. Braunstein, A.K. Pati. 2. Quantum computation over continuous variables S. Lloyd, S.L. Braunstein. 3. Error correction for continuous quantum variables S.L. Braunstein. 4. Deutsch-Jozsa algorithm for continuous variables A.K. Pati, S.L. Braunstein. 5. Hybrid quantum computing S. Lloyd. 6. Efficient classical simulation of continuous variable quantum information processes S.D. Bartlett, B.C. Sanders, S.L. Braunstein, K. Nemoto. Part II: Quantum Entanglement. 7. Introduction to entanglement-based protocols S.L. Braunstein, A.K. Pati. 8. Teleportation of continuous uantum variables S.L. Braunstein, H.J. Kimble. 9. Experimental realization of continuous variable teleportation A. Furusawa, H.J. Kimble. 10. Dense coding for continuous variables S.L. Braunstein, H.J. Kimble. 11. Multipartite Greenberger-Horne-Zeilinger paradoxes for continuous variables S. Massar, S. Pironio. 12. Multipartite entanglement for continuous variables P. van Loock, S.L. Braunstein. 13. Inseparability criterion for continuous variable systems Lu-Ming Duan, G. Giedke, J.I. Cirac, P. Zoller. 14. Separability criterion for Gaussian states R. Simon. 15. Distillability and entanglement purification for Gaussian states G. Giedke, Lu-Ming Duan, J.I. Cirac, P. Zoller. 16. Entanglement purification via entanglement swapping S. Parke, S. Bose, M.B. Plenio. 17. Bound entanglement for continuous variables is a rare phenomenon P. Horodecki, J.I. Cirac, M. Lewenstein. Part III: Continuous Variable Optical-Atomic Interfacing. 18. Atomic continuous variable processing and light-atoms quantum interface A. Kuzmich, E.S. Polzik. Part IV: Limits on Quantum Information and Cryptography. 19. Limitations on discrete quantum information and cryptography S.L. Braunstein, A.K. Pati. 20. Quantum cloning with continuous variables N.J. Cerf. 21. Quantum key distribution with continuous variables in optics T.C. Ralph. 22. Secure quantum key distribution using squeezed states D. Gottesman, J. Preskill. 23. Experimental demonstration of dense coding and quantum cryptography with continuous variables Kunchi Peng, Qing Pan, Jing Zhang, Changde Xie. 24. Quantum solitons in optical fibres: basic requisites for experimental quantum communication G. Leuchs, Ch. Silberhorn, E. Konig, P.K. Lam, A. Sizmann, N. Korolkova. Index.

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Quantum Information with Continuous Variables

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

Entangled quantum states are not separable, regardless of the spatial separation of their components This is a manifestation of an aspect of quantum mechanics known as quantum non-locality An important consequence of this is that the measurement of the state of one particle in a two-particle entangled state defines the state of the second particle instantaneously, whereas neither particle possesses its own well-defined state before the measurement Experimental realizations of entanglement have hitherto been restricted to two-state quantum systems, involving, for example, the two orthogonal polarization states of photons Here we demonstrate entanglement involving the spatial modes of the electromagnetic field carrying orbital angular momentum As these modes can be used to define an infinitely dimensional discrete Hilbert space, this approach provides a practical route to entanglement that involves many orthogonal quantum states, rather than just two Multi-dimensional entangled states could be of considerable importance in the field of quantum information, enabling, for example, more efficient use of communication channels in quantum cryptography

Entanglement of the orbital angular momentum states of photons

How can one prove that a given state is entangled? In this paper we review different methods that have been proposed for entanglement detection. We first explain the basic elements of entanglement theory for two or more particles and then entanglement verification procedures such as Bell inequalities, entanglement witnesses, the determination of nonlinear properties of a quantum state via measurements on several copies, and spin squeezing inequalities. An emphasis is given on the theory and application of entanglement witnesses. We also discuss several experiments, where some of the presented methods have been implemented.

Entanglement detection

Like all of quantum information theory, quantum cryptography is traditionally based on two-level quantum systems. In this paper, a protocol for quantum key distribution based on higher-dimensional systems is presented. An experimental realization using an interferometric setup is also proposed. Analyzing this protocol from the practical side, one finds an increased key creation rate while keeping the initial laser pulse rate constant. Analyzing it for the case of intercept/resend eavesdropping strategy, an increased error rate is found compared to two-dimensional systems, hence an advantage for the legitimate users to detect an eavesdropper.

Quantum Cryptography using larger alphabets

All incoherent as well as 2- and 3-qubit coherent eavesdropping strategies on the six-state protocol of quantum cryptography are classified. For a disturbance of $1/6$, the optimal incoherent eavesdropping strategy reduces to the universal quantum cloning machine. Coherent eavesdropping cannot increase Eve's Shannon information, neither on the entire string of bits, nor on the set of bits received undisturbed by Bob. However, coherent eavesdropping can increase as well Eve's Renyi information as her probability of guessing correctly all bits. The case that Eve delays the measurement of her probe until after the public discussion on error correction and privacy amplification is also considered. It is argued that by doing so, Eve gains only negligibly small additional information.

Incoherent and coherent eavesdropping in the six-state protocol of quantum cryptography

Bell inequality, Bell states and maximally entangled states for n qubits

Non-linear quantum state transformation of spin-1=2

A non-linear quantum state transformation is presented. The transformation, which operates on pairs of spin-1/2, can be used to distinguish optimally between two non-orthogonal states. Similar transformations applied locally on each component of an entangled pair of spin-1/2 can be used to transform a mixed nonlocal state into a quasi-pure maximally entangled singlet state. In both cases the transformation makes use of the basic building block of the quantum computer, namely the quantum-XOR gate.

H. Bechmann-Pasquinucci

Papers

Quantum Cryptography using larger alphabets

Incoherent and coherent eavesdropping in the six-state protocol of quantum cryptography

Bell inequality, Bell states and maximally entangled states for n qubits

Non-linear quantum state transformation of spin-1=2

Nonlinear quantum state transformation of spin-1/2