Unextendible product bases and extremal density matrices with positive partial transpose

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.

23 pages, 2 figures, 1 table. V2: Fixed fig.1 not showing

References in corpus (2)

• Extreme points of the set of density matrices with positive partial transpose
• Numerical studies of entangled PPT states in composite quantum systems

Cited by in corpus (20)

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• Facial structures for various notions of positivity and applications to the theory of entanglement
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• The Minimum Size of Qubit Unextendible Product Bases
• Constructing UMEB from maximally entangled basis
• Unextendible Product Basis for Fermionic Systems
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• Extremal entanglement witnesses
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• Constructing unextendible product bases from multiqubit ones
• Nongeneric positive partial transpose states of rank five in $3\times 3$ dimensions
• Constructions of unextendible entangled bases
• Spectral characterizations of entanglement witnesses