PPT states of almost maximal Schmidt number

Abstract

We construct PPT states on ${\mathbb{C}}^m\otimes {\mathbb{C}}^n$ that have Schmidt number asymptotically approaching the smaller local dimension. More specifically, we construct a PPT state with Schmidt number at least $$\lceil \frac{m+n-((m-n{)}^2+4(m+n-1){)}^{1/2}}{2}\rceil .$$ In the case of equal local dimensions ($m=n$), this becomes $n-\lfloor (2n-1{)}^{1/2}\rfloor$, far exceeding the previous constructions which achieved $n/2+O(1)$. In the case of unequal local dimensions, our result shows that there exists a PPT state on ${\mathbb{C}}^n\otimes {\mathbb{C}}^{3n-4}$ with Schmidt number at least $n-1$.

arXiv comment9 pages