Continuous Variable Quantum Systems Research Articles

Remote State Preparation in Quadrature Basis

We present a theoretical scheme for non-teleportation-based exact deterministic remote state preparation (RSP) in continuous variable (CV) quantum systems. It is an infinite dimensional generalization of RSP of equatorial states. It is capable of preparing states of an ensemble that is parameterized by infinitely many real numbers, i.e. by a real function, while the classical communication cost is only one real number. We show the relation of our scheme to the classical Vernam cipher (one-time pad).

Entanglement between remote continuous-variable quantum systems: Effects of transmission loss

We study the effects of losses on the entanglement created between two separate atomic gases by optical probing and homodyne detection of the transmitted light. The system is well-described in the Gaussian state formulation. Analytical results quantifying the degree of entanglement between the two gases are derived and compared with the entanglement in a pair of light pulses generated by an Einstein-Podolsky-Rosen (EPR) source. For low (high) transmission losses, the highest degree of entanglement is obtained by probing with squeezed (antisqueezed) light. In an asymmetric setup where light is only sent one way through the atomic samples, we find that the logarithmic negativity of entanglement attains a constant value -log{sub 2}(N) with N=1/3, irrespective of the loss along the transmission line.

Fidelities for transformations of unknown quantum states

We present a general theoretical formalism to compute the fidelity of transformations of unknown quantum states, and we apply our theory to Gaussian transformations of continuous variable quantum systems. For the case of a Gaussian distribution of displaced coherent states, the theory is readily tractable by a covariance matrix formalism, and a wider class of states, exemplified by Fock states, can be treated efficiently by the Wigner function formalism. Given the distribution of input states, the optimum feedback gain is identified, and analytical results for the fidelities are presented for recently implemented teleportation and memory storage protocols for continuous variables.

Generating continuous variable quantum codewords in the near-field atomic lithography

Recently, Gottesman et al (2001 Phys. Rev. A 64, 012310) showed how to encode a qubit into a continuous variable quantum system. This encoding was realized by using non-normalizable quantum codewords, which therefore can only be approximated in any real physical set-up. Here we show how a neutral atom, falling through an optical cavity and interacting with a single mode of the intracavity electromagnetic field, can be used to safely encode a qubit into its external degrees of freedom. In fact, the localization induced by a homodyne detection of the cavity field is able to project the near-field atomic motional state into an approximate quantum codeword. The performance of this encoding process is then analysed by evaluating the intrinsic errors induced in the recovery process by the approximated form of the generated codeword.

Schmidt modes and entanglement in continuous-variable quantum systems

The extraction of Schmidt modes for continuous-variable systems is considered. An algorithm based on the singular-value decomposition of a matrix is proposed. It is applied to the entanglement in (i) an atom—photon system with spontaneous emission and (ii) a system of biphotons with spontaneous parametric downconversion (SPDC) of type II. For the atom—photon system, the evolution of entangled states is found to be governed by a parameter approximately equal to the fine-structure constant times the atom-to-electron mass ratio. An analysis is made of the dynamics of atom—photon entanglement on the assumption that the system’s evolution is determined by the superposition of an initial and a final state. It is shown that in the course of emission the entanglement entropy first rises on a timescale of order the excited-state lifetime and then falls, approaching asymptotically a residual level due to the initial energy spread of the atomic packet (momentum spread squared). SPDC of type II is analyzed by means of the polarization density matrix and a newly introduced coherence parameter for two spatially separated modes. The loss of intermodal coherence is addressed that results from the difference in behavior between ordinary-and extraordinary-ray photons in a nonlinear crystal. The degree of intermodal coherence is investigated as a function of the product of crystal length and pump bandwidth.

Quantum cloning

The impossibility of perfectly copying (or cloning) an unknown quantum state is one of the basic rules governing the physics of quantum systems. The processes that perform the optimal approximate cloning have been found in many cases. These ``quantum cloning machines'' are important tools for studying a wide variety of tasks, e.g., state estimation and eavesdropping on quantum cryptography. This paper provides a comprehensive review of quantum cloning machines both for discrete-dimensional and for continuous-variable quantum systems. In addition, it presents the role of cloning in quantum cryptography, the link between optimal cloning and light amplification via stimulated emission, and the experimental demonstrations of optimal quantum cloning.

Partial positive scaling transform: a separability criterion

The problem of constructing a necessary and sufficient condition for establishing the separability of continuous variable systems is revisited. Simon [Phys. Rev. Lett. 84 (2000) 2726] pointed out that such a criterion may be constructed by drawing a parallel between the Peres' partial transpose criterion for finite-dimensional systems and partial time reversal transformation for continuous variable systems. We generalize the partial time reversal transformation to a partial scaling transformation and re-examine the problem using a tomographic description of the continuous variable quantum system. The limits of applicability of the entanglement criteria obtained from partial scaling and partial time reversal are explored.

Correspondence between continuous-variable and discrete quantum systems of arbitrary dimensions

We establish a mapping between a continuous-variable (CV) quantum system and a discrete quantum system of arbitrary dimension. This opens up the general possibility to perform any quantum information task with a CV system as if it were a discrete system. The Einstein-Podolsky-Rosen state is mapped onto the maximally entangled state in any finite-dimensional Hilbert space and thus can be considered as a universal resource of entanglement. An explicit example of the map and a proposal for its experimental realization are discussed.

Covariance measures of intermode correlations and inseparability for continuous variable quantum systems

We introduce several coefficients characterizing the degree of intermode correlations and inseparability for multimode continuous variable quantum systems. All these coefficients are expressed either in terms of traces or determinants of the matrices (or their squares) formed from covariances of the quadrature components, or in terms of partial purities of the modes; thus they are more easy to use in calculations than frequently used entropic measures of entanglement. We compare the new and 'conventional' measures, considering two examples: (1) finite (pure) superpositions of coherent states (including multidimensional even and odd coherent states); (2) the evolution of generic (mixed and squeezed) Gaussian states of two resonantly coupled modes with the most general bilinear coupling in the cases of the nondegenerate parametric amplifier and the parametric converter.

A tomographic approach to quantum nonlocality

We propose a tomographic approach to study quantum nonlocality in continuous variable quantum systems. On one hand we derive a Bell-like inequality for measured tomograms. On the other hand, we introduce pseudospin operators whose statistics can be inferred from the data characterizing the reconstructed state, thus giving the possibility to use standard Bell's inequalities. Illuminating examples are also discussed.

Continuous-variable quantum games

We investigate the quantization of games in which the players can access to a continuous set of classical strategies, making use of continuous-variable quantum systems. For the particular case of the Cournot's duopoly, we find that, even though the two players both act as “selfishly” in the quantum game as they do in the classical game, they are found to virtually cooperate due to the quantum entanglement between them. We also find that the original Einstein–Podolksy–Rosen state contributes to the best profits that the two firms could ever attain. Moreover, we propose a practical experimental setup for the implementation of such quantum games.

Dynamics of entanglement between two trapped atoms

We investigate the dynamics of entanglement between two continuous variable quantum systems. The model system consists of two atoms in a harmonic trap which are interacting by a simplified s-wave scattering. We show, that the dynamically created entanglement changes in a steplike manner. Moreover, we introduce local operators which allow us to violate a Bell-Clauser-Horne-Shimony-Holt inequality adapted to the continuous variable case. The correlations show nonclassical behavior and almost reach the maximal quantum-mechanical value. This is interesting since the states prepared by this interaction are very different from any Einstein-Podolsky-Rosen--like state.

Covariance entanglement measure for two-mode continuous variable systems

We introduce the measure of entanglement of continuous variable quantum systems, expressed in terms of the cross-covariances of the quadrature components or the annihilation/creation operators. Considering the examples of finite and infinite superpositions of coherent and Fock states, we compare the new parameter with the conventional entropic measures of entanglement and with the “linear entropy of entanglement”.

Quantum-information distributors: Quantum network for symmetric and asymmetric cloning in arbitrary dimension and continuous limit

We show that for any Hilbert-space dimension, the optimal universal quantum cloner can be constructed from essentially the same quantum circuit, i.e., we find a universal design for universal cloners. In the case of infinite dimensions (which includes continuous variable quantum systems) the universal cloner reduces to an essentially classical device. More generally, we construct a universal quantum circuit for distributing qudits in any dimension which acts covariantly under generalized displacements and momentum kicks. The behavior of this covariant distributor is controlled by its initial state. We show that suitable choices for this initial state yield both universal cloners and optimized cloners for limited alphabets of states whose states are related by generalized phase-space displacements.