  Finite dimensional entanglement for pure states has been used extensively in
quantum information theory. Depending on the tensor product structure, even set
of separable states can show non-intuitive characters. Two situations are well
studied in the literature, namely the unextendible product basis by Bennett et
al [Phys. Rev. Lett. 82, 5385, (1999)], and completely entangled subspaces
explicitly given by Parthasarathy in [Proc. Indian Acad. Sci. Math. Sci. 114, 4
(2004)]. More recently Boyer, Liss, and Mor [Phys. Rev. A 95, 032308 (2017)];
Boyer and Mor [Preprints 2023080529, (2023)]; and Liss, Mor, and Winter [arXiv:
2309.05144, (2023)] have studied spaces which have only finitely many pure
product states. We carry this further and consider the problem of perturbing
different spaces, such as the orthogonal complement of an unextendible product
basis and also Parthasarathy's completely entangled spaces, by taking linear
spans with specified product vectors. To this end, we develop methods and
theory of variations and perturbations of the linear spans of certain
unextendible product bases, their orthogonal complements, and also
Parthasarathy's completely entangled sub-spaces. Finally we give examples of
perturbations with infinitely many pure product states.
