We discuss the subject of unextendible product bases with the orthogonality condition dropped and we prove that the lowest rank non-separable positive-partial-transpose states, i.e., states of rank 4 in 3 × 3 systems are always locally equivalent to a projection onto the orthogonal complement of a linear subspace spanned by an orthogonal unextendible product basis. The product vectors in the kernels of the states belong to a non-zero measure subset of all general unextendible product bases, nevertheless, they can always be locally transformed to the orthogonal form. This fully confirms the surprising numerical results recently reported by Leinaas et al. Parts of the paper rely heavily on the use of Bezout's theorem from algebraic geometry.
We discuss the subject of unextendible product bases with the orthogonality condition dropped and we prove that the lowest rank non-separable positive-partial-transpose states, i.e., states of rank 4 in 3 × 3 systems are always locally equivalent to a projection onto the orthogonal complement of a linear subspace spanned by an orthogonal unextendible product basis. The product vectors in the kernels of the states belong to a non-zero measure subset of all general unextendible product bases, nevertheless, they can always be locally transformed to the orthogonal form. This fully confirms the surprising numerical results recently reported by Leinaas et al. Parts of the paper rely heavily on the use of Bezout's theorem from algebraic geometry.
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.
Bound-entangled states with positive partial transpose (PPT) can exhibit quantum steering and Bell nonlocality, disproving the long-standing Peres conjecture and its stronger steering counterpart. These counterexamples, obtained through explicit constructions or numerical searches, raise the question of whether they share a common structural origin. We address this question in the low-rank regime by asking whether known counterexamples can be obtained, up to invertible local filtering, from the complement of an unextendible product basis (UPB). We find that the answer depends crucially on the local dimension. In 3⊗3, the rank-four structure of PPT entangled states implies that both the steerable PPT state of Moroder \textit{et al.} and the Bell-nonlocal PPT state of Vértesi and Brunner can be obtained from UPB-complement states under invertible local filtering. For three qubits, in contrast, rank-four states that are PPT across every bipartition are separable across each cut. Moreover, comparison of the product-vector structure of the relevant kernels shows that the known three-qubit Bell-nonlocal PPT states cannot be obtained by invertible local filtering from the complement of any three-qubit UPB. Going beyond rank four, we further show that a known rank-eight three-qutrit state, supported on the complement of a UPB and PPT across every bipartition, is genuinely multipartite entangled. These results show that UPBs provide a unifying structural origin for important low-rank PPT entangled states in 3⊗3, but that nonlocal bound entanglement is not, in general, a UPB phenomenon.
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.
Multiqubit positive-partial-transpose (PPT) entangled states play an important role in quantum information theory. We characterize such states of minimum rank in three-qubit system, namely rank four. Depending on whether the Lorentz invariant is zero, we classify such states into two types. The PPT entangled states constructed by unextendible product bases (UPB) have nonzero invariants, which belong to type I. We provide a method to effectively determine whether a state can be constructed from UPB. For states with zero invariant, which belong to type II, we provide an explicit expression up to equivalence of stochastic local operations and classical communications (SLOCC). It turns out that we can represent them with only one complex parameter. We further study SLOCC-equivalence relation within the expression. We also investigate the Lorentz invariants of multiqubit states with rank less than three and analyze their range.
Deciding whether a mixed quantum state is separable or entangled is a difficult problem in general. Separable states are positive under partial transposition [they are positive-partial-transpose (PPT) states], but this simple test does not exclude all entangled states. In order to understand the entangled PPT states, having so-called bound entanglement, we want to study the extremal PPT states. An extremal PPT state is either a pure product state, then it is separable, or it is entangled, in which case the state and its partial transpose must still be rather low-rank density matrices, although it is known that the rank must be at least 4. In a previous paper we presented a complete classification of the rank-4 entangled PPT states in dimension $3\ifmmode\times\else\texttimes\fi{}3$, generalizing the construction by Bennett et al. from unextendible sets of orthogonal product vectors. In the present paper we continue our investigations of the low-rank entangled PPT states, mostly in dimension $3\ifmmode\times\else\texttimes\fi{}3$, using a combination of analytical and numerical methods. We use perturbation theory in order to construct rank-5 entangled PPT states close to the known rank-4 states, and in order to compute dimensions and study the geometry of surfaces of low-rank PPT states. We exploit the close connection between low-rank PPT states and product vectors. In particular, we show how to reconstruct a low-rank PPT state from a sufficient number of product vectors in its kernel. It may seem surprising that the number of product vectors needed may be smaller than the dimension of the kernel.