040ca38cefb1d9226d79c05dd25469cb-Supplemental.pdf

If there is a bingo on mode-k, the m-th row of the mode-k expansion of P is a constant multiple of the (m 1)-th row, where mis a number determined by the bingo position. When a row is a constant multiple of another row, the rank of the matrix is reduced by a maximum of one, which means Rank(P(k)) Ik 1. In the same way, if there are bk bingos, then bk rows are constant multiple of the other rows, which means Rank(P(k)) Ik bk. For any positive tensor P, rank(P) = 1 if and only if its all many-body θparameters are 0. Proof. First, we show that rank(P) = 1 implies all many-body θ-parameters are 0. From the assumption of rank(P) = 1, the m-th row of the mode-k expansion of P have to be a constant multiple of the (m 1)-th row for all m= {2,...,Ik}and k [d].