A conjecture on the number of independent vector variables in a multiqubit orthogonal product basis

In this paper, we investigate the number of independent vector variables in an n-qubit Orthogonal Product Basis (OPB). Here, the term “independent vector variable” refers to a structural element in the construction of a Unextendible Orthogonal Matrix (UOM). Intuitively, since fixing one state |x in a qubit basis {|x, |x} uniquely determines its orthogonal partner |x , we treat the pair as a single degree of freedom — one variable — rather than two. Based on this counting measure, we propose the conjecture that the number is upper bounded by 2ⁿ − 1. We show that if the number of orthogonal pairs of variables is minimum, then the matrix corresponding to the OPB contains a column consisting of exactly one variable. In this case the conjecture holds. We also demonstrate more OPBs true for the conjecture, when they have a column containing a small number of independent variables.