We construct a parameterized family of $n\otimes n$ PPT (positive partial transpose) states of corank one for each $n\ge 3$. With a suitable choice of parameters, we show that they are $n\otimes n$ PPT entangled edge states of corank one for $3\le n\le 1000$. They violate the range criterion for separability in the most extreme way. Note that corank one is the smallest possible corank for such states. The corank of the partial transpose is given by $2n-3$, which is also the smallest possible corank for the partial transposes of PPT entangled edge states of corank one. They provide the first explicit examples of such states for $n\ge 4$.

中文翻译：

---

具有正部分转置的柯兰克一的纠缠边缘状态

我们为每个 $n\ge 3$ 构建了一个参数化的家庭，其中包含 corank one 的 $n\otimes n$ PPT（正部分转置）状态。通过选择合适的参数，我们表明它们是 $3\le n\le 1000$ 的 corank one 的 $n\otimes n$ PPT 纠缠边缘状态。它们以最极端的方式违反了可分离性的范围标准。请注意，corank one 是此类状态的最小可能 corank。部分转置的corank由$2n-3$给出，这也是corank one的PPT纠缠边缘状态的部分转置的最小可能corank。他们为 $n\ge 4$ 提供了此类状态的第一个显式示例。