Multiqubit UPB: The method of formally orthogonal matrices

TL;DR

This paper introduces a novel method using formal orthogonal matrices to construct unextendible product bases (UPBs) and entangled states in multiqubit systems, expanding the known sizes and properties of such bases.

Contribution

It presents a new construction technique for UPBs of specific sizes and a method for creating multiqubit entangled states with positive partial transposes.

Findings

Constructed many new UPBs of various sizes.

Provided a new construction for UPBs of size n+1 when n ≡ 3 mod 4.

Developed a method for generating multiqubit entangled states with positive partial transposes.

Abstract

We use formal matrices whose entries we view as vector variables taking unit vectors values in one-qubit Hilbert spaces of a multiqubit quantum system. We construct many unextendible product bases (UPBs) of new sizes in such systems and provide a new construction of UPBs of $n$ qubits of cardinality $n+1$ when $n\equiv 3 \pmod{4}$. We also give a new method of constructing multiqubit entangled states with all partial transposes positive.