  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.

Quantum Physics