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arXiv:1210.0111 · Physical Review A

Qubit-qudit states with positive partial transpose

Lin Chen, Dragomir Ž. Ðoković

Published Dec 27, 2012• 34 citations
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Abstract

We show that the length of a qubit-qutrit separable state is equal to $\mathrm{max}(r,s)$, 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.

DOI
10.1103/physreva.86.062332

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We show that the length of a qubit-qutrit separable state is equal to max(r,s)\mathrm{max}(r,s)max(r,s), where rrr is the rank of the state and sss the rank of its partial transpose. We refer to the ordered pair (r,s)(r,s)(r,s) as the birank of this state. We also construct examples of qubit-qutrit separable states of any feasible birank (r,s)(r,s)(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)(d+1,d+1)(d+1,d+1) is equal to d+1d+1d+1. We also show that all qubit-qudit PPT entangled states of birank (d+1,d+1)(d+1,d+1)(d+1,d+1) can be built in a simple way from edge states. If VVV is a subspace of dimension k<dk<dk<d in a 2\ensuremath⨂d2\ensuremath{\bigotimes}d2\ensuremath⨂d space such that VVV contains no product vectors, we show that the set of all product vectors in V\ensuremath⊥{V}^{\ensuremath{\perp}}V\ensuremath⊥ is a vector bundle of rank d\ensuremath−kd\ensuremath{-}kd\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 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.
Whenever EEE 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 EEE 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)(8,6)(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.
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.
One of the most important problems in quantum information is the separability problem, which asks whether a given quantum state is separable. We investigate multipartite states of rank at most 4 which are PPT (i.e., all their partial transposes are positive semidefinite). We show that any PPT state of rank 2 or 3 is separable and has length at most 4. For separable states of rank 4, we show that they have length at most 6. It is six only for some qubit–qutrit or multiqubit states. It turns out that any PPT entangled state of rank 4 is necessarily supported on a 3⊗3 or a 2⊗2⊗2 subsystem. We obtain a very simple criterion for the separability problem of the PPT states of rank at most 4: such a state is entangled if and only if its range contains no product vectors. This criterion can be easily applied since a four-dimensional subspace in the 3⊗3 or 2⊗2⊗2 system contains a product vector if and only if its Plücker coordinates satisfy a homogeneous polynomial equation (the Chow form of the corresponding Segre variety). We have computed an explicit determinantal expression for the Chow form in the former case, while such an expression was already known in the latter case.
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.