低秩密度矩阵可分性的运算准则和构造性检验
Operational criterion and constructive checks for the separability of low rank density matrices
摘要：
本文考虑了Hilbert空间上的低秩密度算子$\varrho$，对于任意的$M$和$N$（$M\leq N$），以及正的部分转置（PPT）$\varrho^{T_A}\ge 0$ .对于秩$r(\varrho) \leq N$，我们证明了有一个PPT是必要的和充分的$\varrho$是可分离的，在这种情况下，我们还提供了其最小分解的纯产品状态。从这个结果可以看出，不存在具有PPT的秩3束缚纠缠态。我们还提出了一个必要和充分条件的一般密度矩阵的$\varrho$和$\varrho^{T_A}$的秩之和满足$r(\varrho)+r(\varrho^{T_A}) \le 2MN-M-N+2$的可分性。这种可分性条件具有构造性检查的形式，因此也为可分离状态提供了纯乘积状态分解，并且它适用于耦合多项式方程组具有有限数量解的情况，正如大多数情况下所预期的那样。
英文摘要：
We consider low rank density operators $\varrho$ supported on a $M\times N$ Hilbert space for arbitrary $M$ and $N$ ($M\leq N$) and with a positive partial transpose (PPT) $\varrho^{T_A}\ge 0$. For rank $r(\varrho) \leq N$ we prove that having a PPT is necessary and sufficient for $\varrho$ to be separable; in this case we also provide its minimal decomposition in terms of pure product states. It follows from this result that there is no rank 3 bound entangled states having a PPT. We also present a necessary and sufficient condition for the separability of generic density matrices for which the sum of the ranks of $\varrho$ and $\varrho^{T_A}$ satisfies $r(\varrho)+r(\varrho^{T_A}) \le 2MN-M-N+2$. This separability condition has the form of a constructive check, providing thus also a pure product state decomposition for separable states, and it works in those cases where a system of couple polynomial equations has a finite number of solutions, as expected in most cases.
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