Tighten the bounds on the number of OPB equivalence classes

Obtain tighter lower and upper bounds on the number a_n of equivalence classes of n-qubit orthogonal product bases, thereby narrowing the gap between the current estimates and achieving a more precise understanding of its growth.

Background

The paper denotes by a_n the number of equivalence classes of n-qubit orthogonal product basis formal matrices under row permutations, column permutations, variable renamings, and direction flips. It establishes the recursive lower bound a_{n+1} >= binom(a_n+1,2) and the Bell-number upper bound a_n <= B_{2^{n-1}}^n, which together imply the doubly exponential asymptotic scale a_n = 2^{2^{n+o(n)}}.

The authors identify improving these estimates as an unresolved problem because the existing lower and upper bounds differ substantially in their detailed growth rates. Sharper bounds would provide a more precise enumeration of essentially different multiqubit orthogonal product bases.