On the rank of extremal marginal states

Let ${\rho }_1$ and ${\rho }_2$ be two states on ${\mathbb{C}}^{d_1}$ and ${\mathbb{C}}^{d_2}$ respectively. The marginal state space, denoted by $\mathcal{C}({\rho }_1,{\rho }_2)$, is the set of all states $\rho$ on ${\mathbb{C}}^{d_1}\otimes {\mathbb{C}}^{d_2}$ with partial traces ${\rho }_1,{\rho }_2$. K. R. Parthasarathy established that if $\rho$ is an extreme point of $\mathcal{C}({\rho }_1,{\rho }_2)$, then the rank of $\rho$ does not exceed $\sqrt{d_1^2+d_2^2-1}$. Rudolph posed a question regarding the tightness of this bound. In 2010, Ohno gave an affirmative answer by providing examples in low-dimensional matrix algebras ${\mathbb{M}}_3$ and ${\mathbb{M}}_4$. This article aims to provide a positive answer to the Rudolph question in various matrix algebras. Our approaches, to obtain the extremal marginal states with tight upper bound, are based on Choi-Jamiołkowski isomorphism and tensor product of extreme points.