  Teleportation is a basic primitive for quantum communication and quantum
computing. We address the problem of continuous-variable (unconditional and
conditional) teleportation of a pure single-photon state and a mixed attenuated
single-photon state generally in a nonunity gain regime. Our figure of merit is
the maximum of negativity of the Wigner function that witnesses highly
non-classical feature of the teleported state. We find that negativity of the
Wigner function of the single-photon state can be {\em unconditionally}
teleported for arbitrarily weak squeezed state used to create the entangled
state shared in the teleportation. In contrast, for the attenuated
single-photon state there is a strict threshold squeezing one has to surpass in
order to successfully teleport the negativity of its Wigner function. The {\em
conditional} teleportation allows to approach perfect transmission of the
single photon for an arbitrarily low squeezing at a cost of a success rate. On
the other hand, for the attenuated single photon conditional teleportation
cannot overcome the squeezing threshold of the unconditional teleportation and
it approaches negativity of the input state only if the squeezing
simultaneously increases. However, as soon as the threshold squeezing is
surpassed the conditional teleportation still pronouncedly outperforms the
unconditional one. The main consequences for quantum communication and quantum
computing with continuous variables are discussed.
