On the Existence of 13×9 Unextendible Orthogonal Matrices

Unextendible product bases correspond to unextendible orthogonal matrices (UOMs), which are a key notion in quantum information theory. We investigate the problem of constructing 13×9 UOMs U13,9. We present the notions of full, maximum and minimum columns to characterize a general U13,9. We show that every column of U13,9 has at least two pairs of orthogonal variables, and U13,9 does not have nine full columns. We also show that the multiplicity of each variable in a column of U13,9 is at most four. When this upper bound is saturated for four variables, it turns out that they respectively have multiplicities (4, 4, 4, 1), (4, 4, 3, 2), (4, 3, 4, 2), or (4, 3, 3, 3). To exclude the existence of U13,9 with any one of them, we present the notion of diagonal-type matrices with four types. None of these types turn out to be a submatrix of U13,9. They imply that no U13,9 has at the same time eight full columns. We further show that every column of U13,9 contains at least three and at most six independent variables. Our results stand for the latest progress on UOMs and help construct U13,9 by programs.