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arXiv:quant-ph/0607087 · Physical Review A

Continuous-variable teleportation in the characteristic-function description

Paulina Marian, Tudor A. Marian

Published Oct 9, 2006• 97 citations
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Abstract

We give a description of the continuous-variable teleportation protocol in terms of the characteristic functions of the quantum states involved. The Braunstein-Kimble protocol is written for an unbalanced homodyne measurement and arbitrary input and resource states. We show that the output of the protocol is a superposition between the input one-mode field and a classical one induced by measurement and classical communication. We choose to describe the input state distortion through teleportation by the average photon number of the measurement-induced field. Only in the case of symmetric Gaussian resource states we find a relation between the optimal added noise and the minimal EPR correlations used to define inseparability.

DOI
10.1103/physreva.74.042306

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We give a description of the continuous-variable teleportation protocol in terms of the characteristic functions of the quantum states involved. The Braunstein-Kimble protocol is written for an unbalanced homodyne measurement and arbitrary input and resource states. We show that the output of the protocol is a superposition between the input one-mode field and a classical one induced by measurement and classical communication. We choose to describe the input state distortion through teleportation by the average photon number of the measurement-induced field. Only in the case of symmetric Gaussian resource states we find a relation between the optimal added noise and the minimal EPR correlations used to define inseparability.
In contrast to discrete-variable teleportation, a quantum state is imperfectly transferred from a sender to a remote receiver in a continuous-variable setting. We recall the ingenious scheme proposed by Braunstein and Kimble for teleporting a one-mode state of the quantum radiation field. By analyzing this protocol, we have previously proven the factorization of the characteristic function of the output state. This indicates that teleportation is a noisy process that alters, to some extent, the input state. Teleportation with a two-mode Gaussian EPR state can be described in terms of the superposition of a distorting field with the input one. Here we analyze the one-mode Gaussian distorting-field state. Some of its most important properties are determined by the statistics of a positive EPR operator in the two-mode Gaussian resource state. We finally examine the fidelity of teleportation of a coherent state when using an arbitrary resource state.
A characteristic function (CF) based formalism for the representation of quantum optical operations pertaining to the Continuous Variable (CV) quantum teleportation protocol for general resource and input states is introduced; allowing for modifications of basic CV teleportation; such as lossy homodyne measurements and the presence of thermal noise. The output state CF is given as a product of the CFs of resource and input. The use of non-Gaussian resources is studied by means of a general class of two-mode squeezed Bell-like states that include as special cases Gaussian, non-Gaussian and "degaussified" resources; it is shown that Bell-like resources optimized ("tailored") for maximum fidelity yield a remarkable improvement in fidelity of teleportation for the studied input states. A further generalization is introduced with two-mode squeezed superpositions of Fock states including finite truncations of Gaussian states as special cases; it is shown that the optimization for maximum fidelity reduces these resources to truncated Gaussian states. Another class of non-Gaussian resources is introduced, optimized squeezed cat-like states; their performance is shown to be higher than that of a Gaussian state; but lower than that for the Bell-like states. It is shown that the optimal non-Gaussian resources are those that most closely realize the simultaneous maximization of the entanglement, the affinity with the two-mode squeezed vacuum and the (suitably measured) amount of non-Gaussianity. The teleportation of coherent state inputs is studied using squeezed Bell-like and squeezed cat-like states superimposed over Gaussian modes representing thermal noise; it is shown that the optimized non-Gaussian resources are more robust in the presence of noise then Gaussian resources.
We analyze the statistics of observables in continuous-variable (CV) quantum teleportation in the formalism of the characteristic function. We derive expressions for average values of output-state observables, in particular, cumulants which are additive in terms of the input state and the resource of teleportation. Working with a general class of teleportation resources, the squeezed-bell-like states, which may be optimized in a free parameter for better teleportation performance, we discuss the relation between resources optimal for fidelity and those optimal for different observable averages. We obtain the values of the free parameter of the squeezed-bell-like states which optimize the central momenta and cumulants up to fourth order. For the cumulants the distortion between in and out states due to teleportation depends only on the resource. We obtain optimal parameters ${\ensuremath{\Delta}}_{(2)}^{\mathrm{opt}}$ and ${\ensuremath{\Delta}}_{(4)}^{\mathrm{opt}}$ for the second- and fourth-order cumulants, which do not depend on the squeezing of the resource. The second-order central momenta, which are equal to the second-order cumulants, and the photon number average are also optimized by the resource with ${\ensuremath{\Delta}}_{(2)}^{\mathrm{opt}}$. We show that the optimal fidelity resource, which has been found previously to depend on the characteristics of input, approaches for high squeezing to the resource that optimizes the second-order momenta. A similar behavior is obtained for the resource that optimizes the photon statistics, which is treated here using the sum of the squared differences in photon probabilities of input versus output states as the distortion measure. This is interpreted naturally to mean that the distortions associated with second-order momenta dominate the behavior of the output state for large squeezing of the resource. Optimal fidelity resources and optimal photon statistics resources are compared, and it is shown that for mixtures of Fock states both resources are equivalent.
Quantum teleportation of an unknown broadband electromagnetic field is investigated. The continuous-variable teleportation protocol by Braunstein and Kimble [Phys. Rev. Lett. 80, 869 (1998)] for teleporting the quantum state of a single mode of the electromagnetic field is generalized for the case of a multimode field with finite bandwith. We discuss criteria for continuous-variable teleportation with various sets of input states and apply them to the teleportation of broadband fields. We first consider as a set of input fields (from which an independent state preparer draws the inputs to be teleported) arbitrary pure Gaussian states with unknown coherent amplitude (squeezed or coherent states). This set of input states, further restricted to an alphabet of coherent states, was used in the experiment by Furusawa et al. [Science 282, 706 (1998)]. It requires unit-gain teleportation for optimizing the teleportation fidelity. In our broadband scheme, the excess noise added through unit-gain teleportation due to the finite degree of the squeezed-state entanglement is just twice the (entanglement) source's squeezing spectrum for its ``quiet quadrature.'' The teleportation of one half of an entangled state (two-mode squeezed vacuum state), i.e., ``entanglement swapping,'' and its verification are optimized under a certain nonunit gain condition. We will also give a broadband description of this continuous-variable entanglement swapping based on the single-mode scheme by van Loock and Braunstein [Phys. Rev. A 61, 10 302 (2000)].
We present a comprehensive investigation of nonideal continuous-variable quantum teleportation implemented with entangled non-Gaussian resources. We discuss in a unified framework the main decoherence mechanisms, including imperfect Bell measurements and propagation of optical fields in lossy fibers, applying the formalism of the characteristic function. By exploiting appropriate displacement strategies, we compute analytically the success probability of teleportation for input coherent states and two classes of non-Gaussian entangled resources: two-mode squeezed Bell-like states (that include as particular cases photon-added and photon-subtracted de-Gaussified states), and two-mode squeezed catlike states. We discuss the optimization procedure on the free parameters of the non-Gaussian resources at fixed values of the squeezing and of the experimental quantities determining the inefficiencies of the nonideal protocol. It is found that non-Gaussian resources enhance significantly the efficiency of teleportation and are more robust against decoherence than the corresponding Gaussian ones. Partial information on the alphabet of input states allows further significant improvement in the performance of the nonideal teleportation protocol.