Unextendible product bases and extremal density matrices with positive partial transpose

TL;DR

This paper explores extending the construction of entangled PPT states using unextendible product bases from 3x3 systems to higher dimensions by relaxing orthogonality conditions, supported by numerical evidence.

Contribution

It proposes a generalization of UPB-based entangled state construction to higher dimensions using non-orthogonal UPBs, supported by numerical studies.

Findings

Numerical evidence of one-parameter families of generalized states in 3x3 systems.

Presence of extremal PPT states of similar form in higher dimensions.

Potential for broader application of UPB constructions beyond orthogonal bases.

Abstract

In bipartite quantum systems of dimension 3x3 entangled states that are positive under partial transposition (PPT) can be constructed with the use of unextendible product bases (UPB). As discussed in a previous publication all the lowest rank entangled PPT states of this system seem to be equivalent, under special linear product transformations, to states that are constructed in this way. Here we consider a possible generalization of the UPB constuction to low-rank entangled PPT states in higher dimensions. The idea is to give up the condition of orthogonality of the product vectors, while keeping the relation between the density matrix and the projection on the subspace defined by the UPB. We examine first this generalization for the 3x3 system where numerical studies indicate that one-parameter families of such generalized states can be found. Similar numerical searches in higher dimensional systems show the presence of extremal PPT states of similar form. Based on these results we suggest that the UPB construction of the lowest rank entangled states in the 3x3 system can be generalized to higher dimensions, with the use of non-orthogonal UPBs.