The degree conjecture for bipartite quantum states which are normalized graph Laplacians was first put forward by Braunstein et al. [Phys. Rev. A 73 (2006) 012320]. The degree criterion, which is equivalent to PPT criterion, is simpler and more efficient to detect the separability of quantum states associated with graphs. Hassan et al. settled the degree conjecture for the separability of multipartite quantum states in [J. Math. Phys. 49 (2008) 0121105]. It is proved that the conjecture is true for pure multipartite quantum states. However, the degree condition is only necessary for separability of a class of quantum mixed states. It does not apply to all mixed states. In this paper, we show that the degree conjecture holds for the mixed quantum states of nearest point graph. As a byproduct, the degree criterion is necessary and sufficient for multipartite separability of N-qubit quantum states associated with graphs.

中文翻译：

---

关于多分量子态可分度猜想的注解

归一化图拉普拉斯算子二部量子态的度数猜想由布劳恩斯坦首先提出等。[物理。牧师 A73(2006) 012320]。度准则相当于 PPT 准则，在检测与图相关的量子态的可分性方面更简单、更有效。哈桑等。解决了[J. 数学。物理。49(2008) 0121105]。证明了该猜想对于纯多分量子态是正确的。然而，度条件只对一类量子混合态的可分性是必要的。它不适用于所有混合状态。在本文中，我们证明度数猜想适用于最近点图的混合量子态。作为副产品，度标准对于多方可分性是必要和充分的ñ- 与图相关的量子位量子态。