  The Hanson-Wright inequality is an upper bound for tails of real quadratic
forms in independent subgaussian random variables. In this work, we extend the
Hanson-Wright inequality for the maximum eigenvalue of the quadratic sum of
random Hermitian tensors under Einstein product. We first prove Weyl inequality
for tensors under Einstein product and apply this fact to separate the
quadratic form of random Hermitian tensors into diagonal sum and coupling
(non-diagonal) sum parts. For the diagonal part, we can apply Bernstein
inequality to bound the tail probability of the maximum eigenvalue of the sum
of independent random Hermitian tensors directly. For coupling sum part, we
have to apply decoupling method first, i.e., decoupling inequality to bound
expressions with dependent random Hermitian tensors with independent random
Hermitian tensors, before applying Bernstein inequality again to bound the tail
probability of the maximum eigenvalue of the coupling sum of independent random
Hermitian tensors. Finally, the Hanson-Wright inequality for the maximum
eigenvalue of the quadratic sum of random Hermitian tensors under Einstein
product can be obtained by the combination of the bound from the diagonal sum
part and the bound from the coupling (non-diagonal) sum part. In Appendix of
this work, we also include the Hanson-Wright inequality under T-product tensor,
which can be derived by the same method of establishing the Hanson-Wright
inequality under Einstein product except changing the rule of tensors product
operation.
