论文标题
持续的张量和多级纠缠转换
Persistent Tensors and Multiqudit Entanglement Transformation
论文作者
论文摘要
我们为新的一类张量构建了张量排名的下限，我们称之为持续张量。我们介绍了三个持续张量的特定家族，其中下限很紧。我们表明，这三个最小级持续张量的家族之间存在一系列变性，可用于研究它们之间的纠缠转换。此外，我们表明，这三个持续张量的家族确实是多QuartiT系统中多Quipit $ \ rm {w} $状态的概括，并且在多Quemdit $ \ rm {ghz} $状态的轨道上均几何闭合。因此，我们表明，可以通过渐近随机的本地操作（SLOCC），从多Quydit $ \ rm {ghz} $状态获得$ \ rm {w} $状态的每一个概括。最后，我们将张量排名所获得的下限扩展到具有持续的汇总和更一般的张量组合的直接总和，我们将其称为块金字塔张量。结果，我们表明，在$ \ rm {ghz} $张量的Kronecker和张量产品下，张量等级是乘法的。
We construct a lower bound of the tensor rank for a new class of tensors, which we call persistent tensors. We present three specific families of persistent tensors, of which the lower bound is tight. We show that there is a chain of degenerations between these three families of minimal-rank persistent tensors that can be used to study the entanglement transformation between them. In addition, we show that these three families of persistent tensors are indeed different generalizations of multiqubit $\rm{W}$ states within multiqudit systems and are geometrically in the orbit closure of multiqudit $\rm{GHZ}$ states. Consequently, we show that one can obtain every one of the generalizations of $\rm{W}$ state from a multiqudit $\rm{GHZ}$ state via asymptotic Stochastic Local Operations and Classical Communication (SLOCC) with rate one. Finally, we extend the obtained lower bound of the tensor rank to direct sums with persistent summands and to even more general combinations of tensors, which we call block pyramidal tensors. As a result, we show that the tensor rank is multiplicative under the Kronecker and tensor products of minimal-rank persistent tensors with the $\rm{GHZ}$ tensor.
