The unextendible product bases of four qubits: Hasse diagrams

Abstract

We consider the unextendible product bases (UPBs) of fixed cardinality $m$ in quantum systems of $n$ qubits. These UPBs are divided into finitely many equivalence classes with respect to an equivalence relation introduced by N. Johnston. There is a natural partial order `$\leq$' on the set of these equivalence classes for fixed $m$, and we use this partial order to study the topological closure of an equivalence class of UPBs. In the case of four qubits, for $m=8,9,10$, we construct explicitly the Hasse diagram of this partial order.