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 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.
We construct a parameterized family of n ⊗ n PPT (positive partial transpose) states of corank one for each n ≥ 3. With a suitable choice of parameters, we show that they are n ⊗ n PPT entangled edge states of corank one for 3 ≤ n ≤ 1000. They violate the range criterion for separability in the most extreme way. Note that corank one is the smallest possible corank for such states. The corank of the partial transpose is given by 2n − 3, which is also the smallest possible corank for the partial transposes of PPT entangled edge states of corank one. They provide the first explicit examples of such states for n ≥ 4.
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.
We construct faces of the convex set of all $2\ensuremath{\bigotimes}4$ bipartite separable states, which are affinely isomorphic to the simplex ${\ensuremath{\Delta}}_{9}$ with 10 extreme points. Every interior point of these faces is a separable state which has a unique decomposition into 10 product states, even though the ranks of the state and its partial transpose are 5 and 7, respectively. We also note that the number 10 is greater than $2\ifmmode\times\else\texttimes\fi{}4$, to disprove a conjecture on the lengths of qubit-qudit separable states. This face is inscribed in the corresponding face of the convex set of all PPT states so that subsimplices ${\ensuremath{\Delta}}_{k}$ of ${\ensuremath{\Delta}}_{9}$ share the boundary if and only if $k\ensuremath{\le}5$. This enables us to find a large class of $2\ensuremath{\bigotimes}4$ PPT entangled edge states with rank 5.