  Non-Gaussian entangled states of light have been found to improve the success
of quantum telepor- tation. Earlier works in the literature focussed mainly on
two-mode non-Gaussian states generated by de-Gaussification of two-mode
squeezed vacuum states. In the current work, we study quan- tum teleportation
with a class of non-Gaussian entangled resource states that are generated at
the output of a passive beam splitter (BS) with different input single mode
non-Gaussian states. In particular, we consider input states that are generated
under successive application of squeezing and photon addition/subtraction
operations in various orders. We focus on identifying what attributes of the
resource states are necessary or sufficient for quantum teleportation (QT). To
this end we first evaluate two attributes considered in the literature, viz.
squeezed vacuum affinity (SVA) and EPR correlation. While SVA is not non-zero
for all two-mode resource states, EPR correlation is neither necessary nor
sufficient of QT. We consider yet another attribute, viz. two-mode quadrature
squeezing as defined by Simon et. al. [Phys. Rev. A 49, 1567 (1994)]. Our
numerical results on the de-Gaussified two-mode squeezed vacuum state as well
as the BS generated non-Gaussian states lead us to the conclusion that two-mode
quadrature squeezing is a necessary condition for QT, in general. We further
demonstrate the plausibility of this conclusion by giving an analytical proof
that two-mode quadrature squeezing is a necessary condition for QT in the case
of symmetric two-mode Gaussian resource states.
