  The Hanson-Wright inequality is an upper bound for tails of real quadratic
forms in independent random variables. In this work, we extend the
Hanson-Wright inequality for the Ky Fan k-norm for the polynomial function of
the quadratic sum of random tensors under Einstein product. We decompose the
quadratic tensors sum into the diagonal part and the coupling part. For the
diagonal part, we can apply the generalized tensor Chernoff bound directly.
But, for the coupling part, we have to apply decoupling method first, i.e.,
decoupling inequality to bound expressions with dependent random tensors with
independent random tensors, before applying generalized tensor Chernoff bound
again to get the the tail probability of the Ky Fan $k$-norm of the coupling
part sum of independent random tensors. At the end, the generalized
Hanson-Wright inequality for the Ky Fan k-norm for the polynomial function of
the quadratic sum of random tensors can be obtained by the combination of the
bound from the diagonal sum part and the bound from the coupling sum part.
