One of the unsolved problems in the characterization of the optimal entanglement witnesses is the existence of optimal witnesses acting on bipartite Hilbert spaces H_{m,n}=C^m\otimes C^n such that the product vectors obeying <e,f|W|e,f>=0 do not span H_{m,n}. So far, the only known examples of such witnesses were found among indecomposable witnesses, one of them being the witness corresponding to the Choi map. However, it remains an open question whether decomposable witnesses exist without the property of spanning. Here we answer this question affirmatively, providing systematic examples of such witnesses. Then, we generalize some of the recently obtained results on the characterization of 2\otimes n optimal decomposable witnesses [R. Augusiak et al., J. Phys. A 44, 212001 (2011)] to finite-dimensional Hilbert spaces H_{m,n} with m,n\geq 3.
We address the problem of optimising entanglement witnesses when a limited fixed set of local measurements can be performed on a bipartite system, thus providing a procedure, feasible also for experiments, to detect entangled states using only the statistics of these local measurements. We completely characterize the class of entanglement witnesses of the form $W = P ^Γ$, where $Γ$ denotes partial transposition, that can be constructed from the measurements of the bipartite operators $σ_{x}\otimesσ_{x},$ $σ_{y}\otimesσ_{y}$ and $σ_{z}\otimesσ_{z}$ in the case of two-qubit systems. In particula…
We construct three-qubit entanglement witnesses with the properties that all the partial transposes have the spanning properties. These witnesses determine faces for separable states whose interior lies in the interior of PPT states, and so they detect sets of PPT entangled states with nonempty interiors. These witnesses also detect all kinds of bi-separable pure product states. Our construction depends on the observation that multi-partite entanglement witnesses correspond to multi-linear maps which are positive.
Two classes of Bell diagonal indecomposable entanglement witnesses in $\mathbb{C}^4 \otimes \mathbb{C}^4$ are considered. Within the first class, we find a generalization of the well-known Choi witness from $\mathbb{C}^3 \otimes \mathbb{C}^3$, while the second one contains the reduction map. Interestingly, contrary to $\mathbb{C}^3 \otimes \mathbb{C}^3$ case, the generalized Choi witnesses are no longer optimal. We perform an optimization procedure of finding spanning vectors, that eventually gives rise to optimal witnesses. Operators from the second class turn out to be optimal, however, with…
Checking the optimality of entanglement witnesses: an application to structural physical approximations
在2008年，最佳纠缠见证者的结构物理近似是可分离态（通常是未归一化的）的猜想被提出。为了证明这一点，在[K。C.哈和S。H. Kye，Separable states with unique decompositions，arXiv：1210.1088v3]，Ha和Kye提出了一种可分解的纠缠见证，其SPA是纠缠的，并论证了它是最优的.本文是在对后一工作[R. Augusiak等人，评论“具有独特分解的可分离态”，arXiv：1304.2040v1]，我们表明，无论是分析和数值，这种纠缠见证不是最佳的，因此它不是一个反例的猜想。为此，我们利用一种方法来检查纠缠证人的最优性已经在[M。Lewenstein等人，A 62，052310（2000）]，然而，迄今为止在文献中几乎没有利用。
We interpret multi-partite genuine entanglement witnesses as simultaneous positivity of various maps arising from them. We apply this result to multi-qubit {\sf X}-shaped Hermitian matrices, and characterize the conditions for them to be genuine entanglement witnesses, in terms of entries. Furthermore, we find all optimal ones among them. They turn out to have the spanning properties, and so they detect non-zero volume set of multi-qubit genuine entanglement. We also characterize decomposability for {\sf X}-shaped entanglement witnesses.
