We investigate conditions on a finite set of multi-partite product vectors for which separable states with corresponding product states have unique decompositions, and show that this is true in most cases if the number of product vectors is sufficiently small. In the three-qubit case, generic five-dimensional spaces give rise to faces of the convex set consisting of all separable states, which are affinely isomorphic to the five-dimensional simplex with six vertices. We construct, as a byproduct, three-qubit entangled positive partial transpose edge states of rank 4 with explicit formulas. This covers those entanglements which cannot be constructed from an unextendible product basis.
We show that the length of a qubit-qutrit separable state is equal to
max(r,s), where
r is the rank of the state and
s the rank of its partial transpose. We refer to the ordered pair
(r,s) as the birank of this state. We also construct examples of qubit-qutrit separable states of any feasible birank
(r,s). We determine the closure of the set of normalized two-qutrit entangled states having positive partial transpose (PPT) of rank 4. The boundary of this set consists of all separable states of length at most 4. We prove that the length of any qubit-qudit separable state of birank
(d+1,d+1) is equal to
d+1. We also show that all qubit-qudit PPT entangled states of birank
(d+1,d+1) can be built in a simple way from edge states. If
V is a subspace of dimension
k<d in a $2\ensuremath{\bigotimes}d$ space such that
V contains no product vectors, we show that the set of all product vectors in ${V}^{\ensuremath{\perp}}$ is a vector bundle of rank $d\ensuremath{-}k$ over the projective line. Finally, we explicitly construct examples of qubit-qudit PPT states (both separable and entangled) of any feasible birank.
Whenever
E is an eight dimensional subspace of the bi-qutrit quantum system whose orthogonal complement is spanned by a vector of Schmidt rank three, we show that there exist PPT entangled edge states with the range space
E whose partial transposes are of rank six, which is the largest possible rank. In this way, we exhibit a huge family of bi-qutrit PPT entangled edge states of type
(8,6). They make faces of the convex set of all PPT states, and we find bi-qutrit PPT entangled edge states of other types on the boundaries of such faces.
We solve the open question of the existence of four-qubit entangled symmetric states with positive partial transpositions (PPT states). We reach this goal with two different approaches. First, we propose a half-analytical--half-numerical method that allows us to construct multipartite PPT entangled symmetric states (PPTESSs) from the qubit-qudit PPT entangled states. Second, we adapt the algorithm allowing us to search for extremal elements in the convex set of bipartite PPT states [Phys. Rev. A 76, 034304 (2007)] to the multipartite scenario. With its aid we search for extremal four-qubit PPTESSs and show that generically they have ranks
(5,7,8). Finally, we provide an exhaustive characterization of these states with respect to their separability properties.
We investigate the geometric structure of the set of spectra of bipartite absolute separable states (
ASEPm,n) and absolute positive partial transpose states (
APPTm,n), \emph{i.e.}, bipartite quantum states that remain separable or PPT, respectively, under all global unitary transformations. First, we establish general geometric properties of absolute convex sets of matrices, their spectra and extreme points. Regarding absolute separability, we present a permutation-symmetric reformulation of the absolute PPT criterion and use it to demonstrate that
APPTm,n is a spectrahedron for all
m≤n: in particular, all its faces are exposed. While
ASEP2,n is a spectrahedron, too, for
ASEPm,n we can in general only show that it is a semialgebraic set for all
m,n≥3. Furthermore, we provide a complete characterization of the faces and extreme points of
APPTm,n and demonstrate that the dimension of a face is determined by the rank of a certain matrix, with maximal proper faces having dimension
mn−m−1.
In the quantitative setting, we provide a rigorous lower bound on the maximal attainable purity of
APPTm,n via an inscribed polytope
Pm,n and conjecture that the maximal purity of
APPTm,n (along with its spectra) coincides with the polytope for arbitrary dimensions except when
m=n=2. Additionally, we also provide a rigorous upper bound on the minimal von Neumann entropy of
APPTm,n and demonstrate numerically that the minimum entropy eventually coincides with the polytope
Pm,n as the local system dimension
n increases. Finally, we show that the relative spectral volume of
APPTm,n decays exponentially in
n by a constant multiplicative factor of the relative volume of the inscribed polytope
Pm,n.