A new geometric representation of qubit and qutrit states based on probability simplexes is used to describe the separability and entanglement properties of density matrices of two qubits. The Peres-Horodecki positive partial transpose (ppt) -criterion and the concurrence inequalities are formulated as the conditions that the introduced probability distributions must satisfy to present entanglement. A four-level system, where one or two states are inaccessible, is considered as an example of applying the elaborated probability approach in an explicit form. The areas of three Triadas of Malevich's squares for entangled states of two qubits are defined through the qutrit state, and the critical values of the sum of their areas are calculated. We always find an interval for the sum of the square areas, which provides the possibility for an experimental checkup of the entanglement of the system in terms of the probabilities.
Entanglement distillation is a key step in quantum information, both theoretically and practically. It has been proven that nonpositive-partial transpose (NPT) entangled states of rank at most four is 1-distillable under local operation and classical communications. In this paper, we investigate the distillation of a more complex family of NPT entangled states, namely, a family of symmetric two-qutrit states [Formula: see text] of rank five with given eigenvectors. We explicitly construct five families of such states by requiring four of the five eigenvalues to be the same. We, respectively, show that some of them are 1-distillable. It turns out that such states may be not 1-distillable for some interval of eigenvalues. We provide some conditions for eigenvalues that allow [Formula: see text] to be 1-distillable or to be 1-undistillable.
Bound entangled (BE) states are strange in nature: non-zero amount of free entanglement is required to create them but no free entanglement can be distilled from them under local operations and classical communication (LOCC). Even though usefulness of such states has been shown in several information processing tasks, there exists no simple method to characterize all these states for an arbitrary composite quantum system. Here we present a
(d−3)/2-parameter family of BE states each with positive partial transpose (PPT). This family of PPT-BE states is introduced by constructing an unextendible product basis (UPB) in
Cd⊗Cd with
d odd and
d≥5. We also provide the `tile structure' corresponding to these UPBs. The range of each such PPT-BE state is contained in a
2(d−1) dimensional entangled subspace whereas the associated UPB-subspace is of dimension
(d−1)2+1. We further show that each of these PPT-BE states can be written as a convex combination of
(d−1)/2 number of rank-4 PPT-BE states. Moreover, we prove that these rank-4 PPT-BE states are extreme points of the convex compact set
P of all PPT states in
Cd⊗Cd, namely the {\it Peres} set. An interesting geometric implication of our result is that the convex hull of these rank-4 PPT-BE extreme points -- the
(d−3)/2-simplex -- is sitting on the boundary between the set
P and the set of non-PPT states. We also discuss consequences of our construction in the context of quantum state discrimination by LOCC.
The 4-qubit unextendible product bases (UPBs) has been recently studied by [Johnston, J. Phys. A: Math. Theor. 47 (2014) 424034]. Based on this study we show that there is only one UPB of size 6 and six UPBs of size 9 in H=C2⊗C2⊗C4, three UPBs of size 9 in K=C4⊗C4, and no UPB of size 7 in H and K. Furthermore we construct a new family of 4-qubit positive-partial-transpose (PPT) entangled states of rank seven, and show that they are also PPT entangled states in H and K, respectively. We analytically derive a geometric measure of a special PPT entangled states.