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arXiv:quant-ph/0401149 · Physical Review Letters

Classical Phase-Space Descriptions of Continuous-Variable Teleportation

Carlton M. Caves, Krzysztof Wódkiewicz

Published Jul 22, 2004• 45 citations
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Abstract

The non-negative Wigner function of all quantum states involved in teleportation of Gaussian states using the standard continuous-variable teleportation protocol means that there is a local realistic phase-space description of the process. This includes the coherent states teleported up to now in experiments. We extend the phase-space description to teleportation of non-Gaussian states using the standard protocol and conclude that teleportation of non-Gaussian pure states with a fidelity of 2/3 is a "gold standard" for this kind of teleportation.

DOI
10.1103/physrevlett.93.040506

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The non-negative Wigner function of all quantum states involved in teleportation of Gaussian states using the standard continuous-variable teleportation protocol means that there is a local realistic phase-space description of the process. This includes the coherent states teleported up to now in experiments. We extend the phase-space description to teleportation of non-Gaussian states using the standard protocol and conclude that teleportation of non-Gaussian pure states with a fidelity of 2/3 is a "gold standard" for this kind of teleportation.
The phase-space method is applied for considering continuous variable quantum teleportation. It is found that the continuous variable quantum teleportation transforms the sss-parametrized phase-space function of an input state into the $(s+\ensuremath{\delta})$-parametrized phase-space function, where the parameter $\ensuremath{\delta}$ is determined by the shared quantum entanglement. It is shown from this result that the Wigner function of the teleported state is always non-negative for ${\mathcal{F}}_{c}\ensuremath{\leqslant}2∕3$ and the Glauber-Sudarshan PPP function non-negative for ${\mathcal{F}}_{c}\ensuremath{\leqslant}1∕2$, where Fc{\mathcal{F}}_{c}Fc​ is the fidelity of the coherent-state teleportation. Furthermore the fidelity between input and output states is calculated when Gaussian states are teleported.
In continuous-variable quantum information, non-Gaussian entangled states that are obtained from Gaussian entangled states via photon subtraction are known to contain more entanglement. This makes them better resources for quantum information processing protocols, such as, quantum teleportation. We discuss the teleportation of non-Gaussian, non-classical Schrödinger-cat states of light using two-mode squeezed vacuum light that is made non-Gaussian via subtraction of a photon from each of the two modes. We consider the experimentally realizable cat states produced by subtracting a photon from the single-mode squeezed vacuum state. We discuss two figures of merit for the teleportation process, (a) the fidelity, and (b) the maximum negativity of the Wigner function at the output. We elucidate how the non-Gaussian entangled resource lowers the requirements on the amount of squeezing necessary to achieve any given fidelity of teleportation, or to achieve negative values of the Wigner function at the output.
Glauber's coherent state is denoted by ∣α⟩\ketα∣α⟩ and its two-mode extension is represented by ∣α,β⟩\ket{α,β}∣α,β⟩. In this work, we introduce a two-mode superposition operator A=tab+ra†b†A=tab+ra^\dagger b^\daggerA=tab+ra†b†, whose action on the two-mode coherent state produces the two-mode coherent superposed quantum state ∣ψ⟩=(tab+ra†b†)∣α,β⟩\ketψ=(tab+ra^\dagger b^\dagger)\ket{α,β}∣ψ⟩=(tab+ra†b†)∣α,β⟩. We investigate the nonclassicality and quantum non-Gaussianity of this state by means of the Wigner distribution and Wigner logarithmic negativity. Once its intrinsic nonclassical and non-Gaussian structure is established, the state is employed as the entangled resource in the Braunstein-Kimble continuous-variable (CV) teleportation protocol. We compute the ideal teleportation fidelity for coherent and squeezed inputs and analyze how the strengths of nonclassicality and non-Gaussianity influence the teleportation efficiency. Our results identify specific parameter regimes where enhanced non-Gaussian features or increased nonclassicality enable fidelities beyond the classical threshold, thereby revealing the operational significance of engineered two-mode quantum states in CV quantum information processing.
Teleportation of optical field states (as continuous quantum variables) is usually described in terms of Wigner functions. This is in marked contrast to the theoretical treatment of teleportation of qubits. In this paper we show that by using the holomorphic representation of the canonical commutation relations, teleportation of continuous quantum variables can be treated in complete analogy to the case of teleportation of qubits. In order to emphasize this analogy, short descriptions of the basic experimental schemes both for teleportation of qubits and of continuous variables are included. We conclude our paper with a brief discussion of the effectiveness of our description of continuous variable teleportation and of the role of localization of quantum states in teleportation problems.
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.
Non-Gaussian operations have been shown to enhance the fidelity of continuous-variable quantum teleportation. However, a disadvantage of these non-Gaussian operations is that they are probabilistic in nature. In this article we study the trade-off between teleportation fidelity and success probability for optimal performance of the teleportation protocol. To this end, we first derive a unified expression for the Wigner characteristic function describing three non-Gaussian states: photon-subtracted, photon-added, and photon-catalyzed two-mode squeezed vacuum states. We then utilize it to obtain the fidelity of teleportation for input coherent and squeezed vacuum states using the aforementioned non-Gaussian resource states. We optimize the product of the relative enhancement in fidelity and the probability of state preparation by tuning the transmissivity of the beam splitters involved in implementing non-Gaussian operations. This leads to a scenario that can be effectively implemented in a laboratory to enhance fidelity. It turns out that among all the non-Gaussian resource states considered, the symmetric one-photon-subtracted two-mode squeezed vacuum state is the most advantageous. We provide the associated optimal squeezing and beam splitter transmissivity values for the non-Gaussian resource states considered, which will be of significant interest to the experimental community. We also consider the effect of imperfect photon detectors on teleportation fidelity. Further, we expect the derived Wigner characteristic function to be useful in state characterization and other quantum information processing protocols.