Quantum versus classical domains for teleportation with continuous variables

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 $\eta >\frac{2}{3}$ 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.