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arXiv:1402.5813 · Journal of Physics A Mathematical and Theoretical

Multi-partite separable states with unique decompositions and construction of three qubit entanglement with positive partial transpose

Kil-Chan Ha, Seung-Hyeok Kye

Published Dec 24, 2014• 10 citations
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Abstract

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.

DOI
10.1088/1751-8113/48/4/045303

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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 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.
Let Sk be the set of separable states on B(Cm⊗Cn) admitting a representation as a convex combination of k pure product states, or fewer. If m>1, n>1, and k⩽max(m,n), we show that Sk admits a subset Vk such that Vk is dense and open in Sk, and such that each state in Vk has a unique decomposition as a convex combination of pure product states, and we describe all possible convex decompositions for a set of separable states that properly contains Vk. In both cases we describe the associated faces of the space of separable states, which in the first case are simplexes, and in the second case are direct convex sums of faces that are isomorphic to state spaces of full matrix algebras. As an application of these results, we characterize all affine automorphisms of the convex set of separable states and all automorphisms of the state space of B(Cm⊗Cn) that preserve entanglement and separability.
We construct three-qubit entanglement witnesses with the full spanning property, by which we mean that all the partial transposes have the spanning properties. These witnesses determine faces for separable states whose interior lies in the interior of positive partial transposes (PPT) states, and so they detect sets of PPT entangled states with nonempty interiors. These witnesses also detect fully bi-separable states as well as all kinds of bi-separable pure product states. Our construction depends on the observation that multi-partite entanglement witnesses correspond to positive multi-linear maps.
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 $2\ensuremath{\bigotimes}d$ space such that VVV 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.
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.
We study separability properties in a five-dimensional set of states of quantum systems composed of three subsystems of equal but arbitrary finite Hilbert space dimension. These are the states that can be written as linear combinations of permutation operators, or equivalently, commute with unitaries of the form $U\ensuremath{\bigotimes}U\ensuremath{\bigotimes}U.$ We compute explicitly the following subsets and their extreme points: (1) triseparable states, which are convex combinations of triple tensor products, (2) biseparable states, which are separable for a twofold partition of the system, and (3) states with positive partial transpose with respect to such a partition. Tripartite entanglement is investigated in terms of the relative entropy of tripartite entanglement and of the trace norm.