Multiqubit orthogonal product bases

Abstract

We use edge-colored complete multigraphs to study complete orthogonal product bases (OPBs) in $n$-qubit systems. We prove that two OPBs are equivalent if and only if their associated multigraphs are isomorphic, thereby reducing OPB classification to graph isomorphism. Within this framework, we establish the upper bound $v\le 2^n-1$ on the number of variables of an $n$-qubit OPB. We also derive $\binom{a_{n-1}+1}{2}\le a_n\le B_{2^{n-1}}^n$ for the number $a_n$ of equivalence classes of $n$-qubit OPBs, where $B_m$ denotes the number of partitions of an $m$-element set. These bounds imply the asymptotic behavior $a_n=2^{2^{n+o(n)}}$. For every OPB, the connectivity pattern of its color layers characterizes local irreducibility, which in turn implies indistinguishability by finite-round local operations and classical communication (LOCC); the existence of a complete color-splitting tree characterizes perfect distinguishability by finite-round LOCC. Finally, we give an algorithm for testing OPB equivalence and a recursive graph algorithm that constructs a finite-round LOCC protocol whenever such perfect discrimination is possible.