This question is tightly related to What separable $\rho$ only admit separable pure decompositions with more than $\mathrm{rank}(\rho)$ terms?. In there, examples were given of separable states $\rho$ with separable decompositions requiring more than $\operatorname{rank}(\rho)$ components. In particular, symmetric Werner states have rank $\binom{n+1}{2}$ but require no less than $n^2$ rank-one separable states to be written.

On the other hand, the standard proof with Carathéodory's theorem sets $\operatorname{rank}(\rho)^2$ as the general upper bound for the sufficient number of elements in a separable pure decomposition of $\rho$.

Is there any example of a state that saturates this upper bound? In other words, a separable state $\rho$ that cannot be decomposed with less than $\operatorname{rank}(\rho)^2$ pure product states?



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