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arXiv:quant-ph/0012001 · Physical Review A

Quantum versus classical domains for teleportation with continuous variables

Samuel L. Braunstein, Christopher A. Fuchs, Harry J. Kimble, Peter van Loock

Published Jul 18, 2001• 136 citations
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Abstract

By considering the utilization of a classical channel without quantum entanglement, fidelity ${F}_{\mathrm{classical}}=\frac{1}{2}$ has been established as setting the boundary between classical and quantum domains in the teleportation of coherent states of the electromagnetic field [S. L. Braunstein, C. A. Fuchs, and H. J. Kimble, J. Mod. Opt. $47,$ 267 (2000)]. We further examine the quantum-classical boundary by investigating questions of entanglement and Bell-inequality violations for the Einstein-Podolsky-Rosen states relevant to continuous variable teleportation. The threshold fidelity for employing entanglement as a quantum resource in teleportation of coherent states is again found to be ${F}_{\mathrm{classical}}=\frac{1}{2}.$ Likewise, violations of local realism onset at this same threshold, with the added requirement of overall efficiency $\ensuremath{\eta}>\frac{2}{3}$ in the unconditional case. By contrast, recently proposed criteria adapted from the literature on quantum-nondemolition detection are shown to be largely unrelated to the questions of entanglement and Bell-inequality violations.

DOI
10.1103/physreva.64.022321

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By considering the utilization of a classical channel without quantum entanglement, fidelity Fclassical=12{F}_{\mathrm{classical}}=\frac{1}{2}Fclassical​=21​ has been established as setting the boundary between classical and quantum domains in the teleportation of coherent states of the electromagnetic field [S. L. Braunstein, C. A. Fuchs, and H. J. Kimble, J. Mod. Opt. 47,47,47, 267 (2000)]. We further examine the quantum-classical boundary by investigating questions of entanglement and Bell-inequality violations for the Einstein-Podolsky-Rosen states relevant to continuous variable teleportation. The threshold fidelity for employing entanglement as a quantum resource in teleportation of coherent states is again found to be Fclassical=12.{F}_{\mathrm{classical}}=\frac{1}{2}.Fclassical​=21​. Likewise, violations of local realism onset at this same threshold, with the added requirement of overall efficiency \ensuremathη>23\ensuremath{\eta}>\frac{2}{3}\ensuremathη>32​ in the unconditional case. By contrast, recently proposed criteria adapted from the literature on quantum-nondemolition detection are shown to be largely unrelated to the questions of entanglement and Bell-inequality violations.
Summary form only given. Traditionally quantum information research has centred on discrete quantum variables, particularly binary variables (qubits). However, recently much attention has focused on quantum information manipulations with continuous variables such as the amplitude and phase of the electromagnetic field. As well as offering the possibility of high data rates, continuous variables also exhibit greater versatility in realizing certain operations. For example teleportation, in which quantum information is passed intact through a classical channel with the assistance of shared entanglement, can be achieved more efficiently using continuous variable techniques. I review recent work on the characterization of teleportation and entanglement swapping and present new work on interferometric tests of teleportation. These interferometric tests are important because they allow the efficacy of teleportation to be checked without knowledge of the input quantum state. They can be applied equally to the cases of discrete and continuous variable inputs and in so doing highlight unexpected similarities. I contrast the various classical limits which emerge from the different ways of assessing teleportation. An obvious omission from the list of continuous variable quantum information manipulations is any proposal for direct analogues of 2 photon interference effects, especially those that lead to Bell inequality type tests of local realism. Bell-type correlations are the defining feature of discrete variable entanglement experimentally. I discuss new work that shows how entanglement of the Bell-type can be observed in continuous variable, homodyne measurement schemes.
The need for high-fidelity quantum teleportation arises in a variety of quantum algorithms and protocols. Unfortunately, conventional continuous-variable teleportation schemes rely on Einstein–Podolsky–Rosen states that yield a fidelity that approaches unity only in the limit of an unphysical amount of squeezing. A new method that utilizes an ensemble of single photon entangled states, qubits, to teleport continuous variable (CV) states with fidelity approaching unity with finite resources was recently proposed by Andersen and Ralph [Phys. Rev. Lett.111, 050504 (2013)]. We extend these ideas to consider the general case of using maximally entangled d-level states, qudits, to teleport a CV state and discuss how the corresponding results are affected. In particular, we find that, by using qudits with dimension greater than two, we can achieve a higher fidelity with comparable resources.
We derive an experimentally testable criterion for the teleportation of quantum states of continuous variables. This criterion is especially relevant to the recent experiment of Furusawa et al. [Science 282, 706-709 (1998)] where an input-output fidelity of 0.58pm0.020.58 \\pm 0.020.58pm0.02 was achieved for optical coherent states. Our derivation demonstrates that fidelities greater than 1/2 could not have been achieved through the use of a classical channel alone; quantum entanglement was a crucial ingredient in the experiment.
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.