In this paper we describe a connection between unextendable product bases (UPB) and positive (P) maps which are not completely positive (CP). We show that inner automorphisms of the set of P maps which are not CP produce extremal extensions of these maps that help in entanglement detection. By constructing such an extension of the well-known Choi map, we strengthen its power to unearth positive under partial transpose entangled states. We further show that the class of maps generated from the Choi map via an inner automorphism naturally detects the entanglement of states in the orthogonal complement of certain UPB. This brings out a hitherto undiscovered connection between the Choi map and UPB. We also show that certain other recently considered extremal extensions are obtainable by such extensions of the Choi map.
In this paper we describe a connection between unextendable product bases (UPB) and positive (P) maps which are not completely positive (CP). We show that inner automorphisms of the set of P maps which are not CP produce extremal extensions of these maps that help in entanglement detection. By constructing such an extension of the well-known Choi map, we strengthen its power to unearth positive under partial transpose entangled states. We further show that the class of maps generated from the Choi map via an inner automorphism naturally detects the entanglement of states in the orthogonal complement of certain UPB. This brings out a hitherto undiscovered connection between the Choi map and UPB. We also show that certain other recently considered extremal extensions are obtainable by such extensions of the Choi map.
In this paper, we discuss extremal extensions of entanglement witnesses based on Choi's map. The constructions are based on a generalization of the Choi map, from which we construct entanglement witnesses. These extremal extensions are powerful in terms of their capacity to detect entanglement of positive under partial transpose (PPT) entangled states and lead to unearthing of entanglement of new PPT states. We also use the Cholesky-like decomposition to construct entangled states which are revealed by these extremal entanglement witnesses.
In bipartite quantum systems of dimension $3\ifmmode\times\else\texttimes\fi{}3$, entangled states that are positive under partial transposition (PPT) can be constructed with the use of unextendible product bases (UPBs). As discussed in a previous publication, all the lowest rank entangled PPT states of this system seem to be equivalent, under $\mathrm{SL}\ensuremath{\bigotimes}\mathrm{SL}$ 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 $3\ifmmode\times\else\texttimes\fi{}3$ 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 $3\ifmmode\times\else\texttimes\fi{}3$ system can be generalized to higher dimensions, with the use of nonorthogonal UPBs.
It is known how to construct, in a bipartite quantum system, a unique low-rank entangled mixed state with positive partial transpose (a PPT state) from an unextendible product basis (UPB), defined as an unextendible set of orthogonal product vectors. We point out that a state constructed in this way belongs to a continuous family of entangled PPT states of the same rank, all related by nonsingular unitary or nonunitary product transformations. The characteristic property of a state $\ensuremath{\rho}$ in such a family is that its kernel $\text{Ker} \ensuremath{\rho}$ has a generalized UPB, a basis of product vectors, not necessarily orthogonal, with no product vector in $\mathrm{Im} \ensuremath{\rho}$, the orthogonal complement of $\text{Ker} \ensuremath{\rho}$. The generalized UPB in $\text{Ker} \ensuremath{\rho}$ has the special property that it can be transformed to orthogonal form by a product transformation. In the case of a system of dimension $3\ifmmode\times\else\texttimes\fi{}3$, we give a complete parametrization of orthogonal UPBs. This is then a parametrization of families of rank 4 entangled (and extremal) PPT states, and we present strong numerical evidence that it is a complete classification of such states. We speculate that the lowest rank entangled and extremal PPT states also in higher dimensions are related to generalized, nonorthogonal UPBs in similar ways.
In bipartite quantum systems of dimension 3× 3 the lowest rank entangled states that are positive under partial transposition (PPT) can be related, by product transformations, to states that are constructed by the use of unextendible product bases (UPB) in the Hilbert space of the system. Here we consider a possible generalization of the product bases, which can be related in a similar way to low-rank entangled density matrices 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 associated UPB. We examine first this generalization for the 3 × 3 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 the same form. Based on these results we indicate that the UPB construction of the lowest rank entangled states in the 3×3 system can be generalized to higher dimensions, with the use of non-orthogonal UPBs of the suggested form.
It is known that some two-qutrit entangled states of rank 4 with positive partial transpose can be built from the unextendible product bases (UPB) [C. H. Bennett, D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin, and B. M. Terhal, Phys. Rev. Lett. 82, 5385 (1999)]. We show that this fact is indeed universal, namely, all such states can be constructed from UPB as conjectured recently by Leinaas, Myrheim, and Sollid. We also classify the five-dimensional subspaces of two qutrits which contain only finitely many product states (up to scalar multiple), and in particular those spanned by an UPB.
We consider entanglement witnesses arising from positive linear maps which generate exposed extremal rays. We show that every entangled state can be detected by one of these witnesses, and this witness detects a unique set of entangled states among those. Therefore, they provide a minimal set of witnesses to detect any kind of entanglement in a sense. Furthermore, if those maps are indecomposable then they detect large classes of entangled states with positive partial transposes which have nonempty relative interiors in the cone generated by all PPT states. We also provide a one-parameter family of indecomposable positive linear maps which generate exposed extremal rays. This gives the first examples of such maps in three-dimensional matrix algebra.