  It is a topic of fundamental and practical importance how a quantum
correlated state can be reliably distributed through a noisy channel for
quantum information processing. The concept of quantum steering recently
defined in a rigorous manner is relevant to study it under certain
circumstances and we here address quantum steerability of Gaussian states to
this aim. In particular, we attempt to reformulate the criterion for Gaussian
steering in terms of local and global purities and show that it is sufficient
and necessary for the case of steering a 1-mode system by a $N$-mode system. It
subsequently enables us to reinforce a strong monogamy relation under which
only one party can steer a local system of 1-mode. Moreover, we show that only
a negative partial-transpose state can manifest quantum steerability by
Gaussian measurements in relation to the Peres conjecture. We also discuss our
formulation for the case of distributing a two-mode squeezed state via one-way
quantum channels making dissipation and amplification effects, respectively.
Finally, we extend our approach to include non-Gaussian measurements, more
precisely, all orders of higher-order squeezing measurements, and find that
this broad set of non-Gaussian measurements is not useful to demonstrate
steering for Gaussian states beyond Gaussian measurements.
