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

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.

significantly revised, title changed, heterogeneous systems considered

References in corpus (7)

• Generic local distinguishability and completely entangled subspaces
• Numerical studies of entangled PPT states in composite quantum systems
• Strong quantum nonlocality for unextendible product bases in heterogeneous systems
• Nonexistence of $n$-qubit unextendible product bases of size $2^n-5$
• Simple sufficient condition for subspace to be completely or genuinely entangled
• Constructing unextendible product bases from the old ones
• Construction of genuinely entangled multipartite subspaces from bipartite ones by reducing the total number of separated parties

Cited by in corpus (3)

• Universal construction of genuinely entangled subspaces of any size
• Unextendible and uncompletable product bases in every bipartition
• Progress in the study of the (non)existence of genuinely unextendible product bases