The unextendible product bases of four qubits: Hasse diagrams

TL;DR

This paper investigates the structure of unextendible product bases in four-qubit systems, using Hasse diagrams to represent the partial order of their equivalence classes for specific sizes, revealing their topological relationships.

Contribution

It explicitly constructs Hasse diagrams for the partial order of UPB classes in four-qubit systems at sizes 8, 9, and 10, advancing understanding of their topological structure.

Findings

Explicit Hasse diagrams for UPB classes at m=8,9,10

Topological closure relations among UPB classes

Classification of UPBs into finitely many equivalence classes

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.