Useful variants and perturbations of completely entangled subspaces and spans of unextendible product bases

TL;DR

This paper investigates how perturbations of unextendible product bases and completely entangled subspaces affect their structure, providing methods and examples that reveal the emergence of infinitely many pure product states.

Contribution

It introduces new methods for analyzing perturbations of entangled subspaces and unextendible product bases, expanding understanding of their stability and structure.

Findings

Perturbations can introduce infinitely many pure product states.

Methods for analyzing variations of entangled subspaces are developed.

Examples demonstrate the effects of perturbations on these spaces.

Abstract

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 [Lett. Math. Phys, 114, 86 (2024)] 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.