Completely positive biquadratic tensors

In this paper, we systemically introduce completely positive biquadratic (CPBQ) tensors and copositive biquadratic tensors. We show that all weakly CPBQ tensors are sum of squares tensors, the CPBQ tensor cone and the copositive biquadratic tensor cone are dual cone to each other. We also show that the outer product of two completely positive matrices is a CPBQ tensor, and the outer product of two copositive matrices is a copositive biquadratic tensor. We then study two easily checkable subclasses of CPBQ tensors, namely positive biquadratic Cauchy tensors and biquadratic Pascal tensors. We show that a biquadratic Pascal tensor is both strongly CPBQ and positive definite.