From unextendible product bases to genuinely entangled subspaces

TL;DR

This paper explores the relationship between unextendible product bases and genuinely entangled subspaces, providing new classes of nonorthogonal UPBs that generate GESs and enabling the construction of genuinely entangled states in multipartite quantum systems.

Contribution

It introduces classes of nonorthogonal UPBs that lead to GESs across various multipartite dimensions and offers a straightforward method to construct genuinely entangled states.

Findings

New classes of nonorthogonal UPBs for any number of parties

Connection established between UPBs and GESs

Method for constructing genuinely entangled states

Abstract

Unextendible product bases (UPBs) are interesting mathematical objects arising in composite Hilbert spaces that have found various applications in quantum information theory, for instance in a construction of bound entangled states or Bell inequalities without quantum violation. They are closely related to another important notion, completely entangled subspaces (CESs), which are those that do not contain any fully separable pure state. Among CESs one finds a class of subspaces in which all vectors are not only entangled, but are genuinely entangled. Here we explore the connection between UPBs and such genuinely entangled subspaces (GESs) and provide classes of nonorthogonal UPBs that lead to GESs for any number of parties and local dimensions. We then show how these subspaces can be immediately utilized for a simple general construction of genuinely entangled states in any such multipartite scenario.