On the tightness of the Parthasarathy and Schmidt rank bounds for extremal tracial couplings
Let $2 \le a \le b$, and let $\rho_A, \rho_B$ be density operators on $\C^a$ and $\C^b$, respectively. Let $\mathcal{C}(\rho_A, \rho_B)$ denote the set of bipartite couplings on $\C^a \otimes \C^b$ with these marginals. Parthasarathy showed that every extreme coupling has rank at most $\KRP(a,b) := \lfloor\sqrt{a^2+b^2-1}\rfloor$. It is not known for which dimension pairs $(a, b)$ this bound is tight. We restrict attention to the tracial case $\rho_A = I_a/a$, $\rho_B = I_b/b$. For the Parthasarathy bound, we construct extreme couplings of rank $r$ from pairs of length-$r$ finite unit-norm…
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