  A spin-$j$ state can be represented by a symmetric tensor of order $N=2j$ and
dimension $4$. Here, $j$ can be a positive integer, which corresponds to a
boson; $j$ can also be a positive half-integer, which corresponds to a fermion.
In this paper, we introduce regularly decomposable tensors and show that a
spin-$j$ state is classical if and only if its representing tensor is a
regularly decomposable tensor. In the even-order case, a regularly decomposable
tensor is a completely decomposable tensor but not vice versa; a completely
decomposable tensors is a sum-of-squares (SOS) tensor but not vice versa; an
SOS tensor is a positive semi-definite (PSD) tensor but not vice versa. In the
odd-order case, the first row tensor of a regularly decomposable tensor is
regularly decomposable and its other row tensors are induced by the regular
decomposition of its first row tensor. We also show that complete
decomposability and regular decomposability are invariant under orthogonal
transformations, and that the completely decomposable tensor cone and the
regularly decomposable tensor cone are closed convex cones. Furthermore, in the
even-order case, the completely decomposable tensor cone and the PSD tensor
cone are dual to each other. The Hadamard product of two completely
decomposable tensors is still a completely decomposable tensor. Since one may
apply the positive semi-definite programming algorithm to detect whether a
symmetric tensor is an SOS tensor or not, this gives a checkable necessary
condition for classicality of a spin-$j$ state. Further research issues on
regularly decomposable tensors are also raised.
