  No-cloning theorem, a profound fundamental principle of quantum mechanics,
also provides a crucial practical basis for secure quantum communication. The
security of communication can be ultimately guaranteed if the output fidelity
via communication channel is above the no-cloning bound (NCB). In quantum
communications using continuous-variable (CV) systems, Gaussian states, more
specifically, coherent states have been widely studied as inputs, but less is
known for non-Gaussian states. We aim at exploring quantum communication
covering CV states comprehensively with distinct sets of unknown states
properly defined. Our main results here are (i) to establish the NCB for a
broad class of quantum non-Gaussian states including Fock states, their
superpositions and Schrodinger-cat states and (ii) to examine the relation
between NCB and quantum non-Gaussianity (QNG). We find that NCB typically
decreases with QNG. Remarkably, this does not mean that quantum non-Gaussian
states are less demanding for secure communication. By extending our study to
mixed-state inputs, we demonstrate that QNG specifically in terms of Wigner
negativity requires more resources to achieve output fidelity above NCB in CV
teleportation. The more non-Gaussian, the harder to achieve secure
communication, which can have crucial implications for CV quantum
communications.
