  Multipartite Einstein-Podolsky-Rosen (EPR) steering and multimode quantum
squeezing are essential resources for various quantum applications. The paper
focuses on studying a coupled three-mode squeezed vacuum (C3MSV), which is a
typical multimode squeezed Gaussian state and will exhibit peculiar steering
property. Using the technique of integration within ordered products, we give
the normal-ordering form for the coupled three-mode squeezing operator and
derive the general analytical expressions of the statistical quantities for the
C3MSV. Under Gaussian measurements, we analyze all bipartite Gaussian steerings
(including no steering, one-way steering and two-way steering) in details and
study the monogamy relations for the C3MSV. Then, we study the decoherence of
all these steerings in noisy channels and find that sudden death will happen in
a certain threshold time. Through the steerings shared in the C3MSV, we propose
conceptual (and ideal) schemes of remotely generating Wigner negativity (WN) by
performing appropriate photon subtraction(s) in the local position. Our
obtained results may lay a solid theoretical foundation for a future practical
study. We also believe that the C3MSV will be one of good candidate resources
in future quantum protocols.
