Quantum information is a rapidly advancing area of interdisciplinary research. It may lead to real-world applications for communication and computation unavailable without the exploitation of quantum properties such as nonorthogonality or entanglement. We review the progress in quantum information based on continuous quantum variables, with emphasis on quantum optical implementations in terms of the quadrature amplitudes of the electromagnetic field.

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Quantum teleportation of optical coherent states was demonstrated experimentally using squeezed-state entanglement. The quantum nature of the achieved teleportation was verified by the experimentally determined fidelity Fexp = 0.58 +/- 0.02, which describes the match between input and output states. A fidelity greater than 0.5 is not possible for coherent states without the use of entanglement. This is the first realization of unconditional quantum teleportation where every state entering the device is actually teleported.

Quantum teleportation is analyzed for states of dynamical variables with continuous spectra, in contrast to previous work with discrete (spin) variables. The entanglement fidelity of the scheme is computed, including the roles of finite quantum correlation and nonideal detection efficiency. A protocol is presented for teleporting the wave function of a single mode of the electromagnetic field with high fidelity using squeezed-state entanglement and current experimental capability.[ S0031-9007(97)

This paper provides necessary and sufficient conditions for constructing a universal quantum computer over continuous variables. As an example, it is shown how a universal quantum computer for the amplitudes of the electromagnetic field might be constructed using simple linear devices such as beam-splitters and phase shifters, together with squeezers and nonlinear devices such as Kerr-effect fibers and atoms in optical cavities. Such a device could in principle perform 'quantum floating point' computations.Problems of noise, finite precision, and error correction are discussed.Quantum computation has traditionally concerned itself with the manipulation of discrete systems such as quantum bits, or 'qubits'.1−2 Many quantum variables, such as position and momentum, or the amplitudes of electromagnetic fields, are continuous. Although noise and finite precision make precise manipulations of continuous variables intrinsically more difficult than the manipulation of discrete variables, because of the recent developments in quantum error correction 3−5 and quantum teleportation 6−7 of continuous quantum variables it is worthwhile addressing the question of quantum computation over continuous variables.At first it might seem that quantum computation over continuous variables is an ill-defined concept. First consider quantum computation over discrete variables. A universal quantum computer over discrete variables such as qubits can be defined to be a 1

The quantum-mechanical framework in which observables are associated with Hermitian operators is too narrow to discuss measurements of such important physical quantities as elapsed time or harmonic-oscillator phase. We introduce a broader framework that allows us to derive quantum-mechanical limits on the precision to which a parameter e.g., elapsed time may be determined via arbitrary data analysis of arbitrary measurements on N identically prepared quantum systems. The limits are expressed as generalized Mandelstam Tamm uncertainty relations, which involve the operator that generates displacements of the parameter e.g., the Hamiltonian operator in the case of elapsed time. This approach avoids entirely the problem of associating a Hermitian operator with the parameter. We illustrate the general formalism, first, with nonrelativistic uncertainty relations for spatial displacement and momentum, harmonic-oscillator phase and number of quanta, and time and energy and, second, with Lorentz-invariant uncertainty relations involving the displacement and Lorentzrotation parameters of the Poincare group.

Quantum teleportation is one of the most important protocols in quantum information. By exploiting the physical resource of entanglement, quantum teleportation serves as a key primitive in a variety of quantum information tasks and represents an important building block for quantum technologies, with a pivotal role in the continuing progress of quantum communication, quantum computing and quantum networks. Here we review the basic theoretical ideas behind quantum teleportation and its variant protocols. We focus on the main experiments, together with the technical advantages and disadvantages associated with the use of the various technologies, from photonic qubits and optical modes to atomic ensembles, trapped atoms, and solid-state systems. Analysing the current state-of-the-art, wefinish by discussing open issues, challenges and potential future implementations. From Science Fiction to RealityIt has been over two decades since the discovery of quantum teleportation, in what is arguably one of the most interesting and exciting implications of the 'weirdness' of quantum mechanics. Previous to this landmark discovery, this fascinating idea belonged to the realm of science fiction. First coined in a 1931 book by Charles H. Fort 1 , the term teleportation has since been used to refer to the process by which bodies and objects are transferred from one location to another, without actually making the journey along the way. Since then it has become a fixture of pop culture, perhaps best exemplified by Star Trek's celebrated catchphrase "Beam me up, Scotty."In 1993, a seminal paper 2 described a quantum information protocol, dubbed quantum teleportation, that shares several of the above features. In this protocol, an unknown quantum state of a physical system is measured and subsequently reconstructed or 'reassembled' at a remote location (the physical constituents of the original system remain at the sending location). This process requires classical communication and excludes superluminal communication. Most importantly, it requires the resource of quantum entanglement 3,4 . Indeed, quantum teleportation can be seen as the protocol in quantum information that most clearly demonstrates the character of quantum entanglement as a resource: Without its presence, such a quantum state transfer would not be possible within the laws of quantum mechanics.Quantum teleportation plays an active role in the progress of quantum information science [5][6][7][8] . On the one hand it is a conceptual protocol crucial in the development of formal quantum information theory, on the other it represents a fundamental ingredient to the development of many quantum technologies. Schemes such as quantum repeaters 9 -pivotal for quantum communication over large distances -quantum gate teleportation 10 , measurement-based computing 11 , and port-based teleportation 12 all derive from the basic scheme. The vision of a quantum network 13 draws inspiration from it. Teleportation has also been used as a simple tool for exploring 'extreme' physics,...

We give a constructive proof that all mixed states of N qubits in a sufficiently small neighborhood of the maximally mixed state are separable. The construction provides an explicit representation of any such state as a mixture of product states. We give upper and lower bounds on the size of the neighborhood, which show that its extent decreases exponentially with the number of qubits. We also discuss the implications of the bounds for NMR quantum computing.