Negative result about the construction of genuinely entangled subspaces from unextendible product bases

TL;DR

This paper proves that genuinely entangled subspaces cannot be constructed from unextendible product bases for many sizes and scenarios, resolving a long-standing open question in quantum information theory.

Contribution

It demonstrates the non-existence of genuinely unextendible UPBs for various sizes and multipartite settings, especially in equal local dimensions.

Findings

No genuinely unextendible UPBs for certain sizes in multipartite systems.

Forbidden cardinalities for UPBs including those related to GESs.

Genuinely entangled subspaces cannot be derived from UPBs in these cases.

Abstract

Unextendible product bases (UPBs) provide a versatile tool with various applications across different areas of quantum information theory. Their comprehensive characterization is thus of great importance and has been a subject of vital interest for over two decades now. An open question asks about the existence of UPBs, which are genuinely unextendible, i.e., they are not extendible even with biproduct vectors. In other words, the problem is to verify whether there exist genuinely entangled subspaces (GESs), subspaces composed solely of genuinely multiparty entangled states, complementary to UPBs. We solve this problem in the negative for many sizes of UPBs in different multipartite scenarios. In particular, in the all-important case of equal local dimensions, we show that there are always forbidden cardinalities for such UPBs, including the minimal ones corresponding to GESs of the maximal dimensions.