Useful variants and perturbations of completely entangled subspaces and spans of unextendible product bases
Ritabrata Sengupta, Ajit Iqbal Singh
TL;DR
This work develops a unified perturbation framework for completely entangled and quasi‑completely entangled subspaces arising from two classic constructions: UPB orthogonal complements and Parthasarathy's van der Monde–based CES. By taking linear spans with carefully chosen product vectors, the authors construct and analyze quasi‑CES (QCES) in multipartite settings, providing explicit examples (notably TILES UPB and tripartite qubits) and showing how perturbations can yield spaces with a finite product-vector content or, in many cases, infinitely many product vectors. The polynomial representation is leveraged to identify product vectors in perturbed spans, yielding exact counts and forms of product states for several configurations, including multipartite qubits and bipartite setups. The results emphasize the nuanced behavior under perturbations: some constructions remain QCES with a fixed product index, while double perturbations frequently generate families with infinitely many product vectors, highlighting rich structure and new directions for classifying entangled subspaces in finite dimensions.
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.