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

$\displaystyle\left\lceil \frac{m + n - \sqrt{(m - n)^2 + 4(m + n - 1)}}{2} \right\rceil.$
In the case of equal local dimensions (m = n), this becomes $n - \lfloor\sqrt{2n - 1}\rfloor$ , far exceeding 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.
Authors:

Nathaniel Johnston
Benjamin Lovitz

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Preprint from arXiv:2609.11849 [quant-ph]

Cite as:

N. Johnston and B. Lovitz. PPT states of almost maximal Schmidt number. E-print: arXiv:2609.11849 [quant-ph], 2026.