Universal construction of genuinely entangled subspaces of any size

TL;DR

This paper presents a universal method to construct genuinely entangled subspaces of any size and for any number of parties, enabling the creation of complex entangled states with broad applications.

Contribution

It introduces a simple, explicit construction of genuinely entangled subspaces using nonorthogonal product bases derived from totally nonsingular matrices, applicable to any multipartite system.

Findings

Constructed genuinely entangled subspaces of arbitrary dimension

Enabled creation of genuinely entangled mixed states with maximal rank

Provided explicit basis for these subspaces

Abstract

We put forward a simple construction of genuinely entangled subspaces -- subspaces supporting only genuinely multipartite entangled states -- of any permissible dimensionality for any number of parties and local dimensions. The method uses nonorthogonal product bases, which are built from totally nonsingular matrices with a certain structure. We give an explicit basis for the constructed subspaces. An immediate consequence of our result is the possibility of constructing in the general multiparty scenario genuinely multiparty entangled mixed states with ranks up to the maximal dimension of a genuinely entangled subspace.