math.OA — Operator Algebras

On the tightness of the Parthasarathy and Schmidt rank bounds for extremal tracial couplings

Contributed by Dipankar Dey, Somnath Jana, Soumyashant Nayak

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 tight frames (FUNTFs) in $\C^a$ and $\C^b$ satisfying a transversality condition. This yields an extreme coupling of rank $b$ for every $2 \le a < b$, and hence tightness of $\KRP(a,b)$ whenever $a<b$ and $b\ge(a^2-1)/2$; in particular, along the whole qubit-qudit line $(2,b)$, $b\ge3$. This recovers the two previously known tracial cases with $a<b$, namely $(2,3)$ and $(3,4)$. The Schmidt decomposition and subadditivity of rank give the lower bound $\ES(a, b) := \lceil b/a\rceil$ for the rank of every tracial coupling, extreme or not. For the Schmidt bound, we use sparse Kraus operators built from weighted partial permutation matrices. Setting $r := \ES(a, b)$, we construct an extreme coupling of rank $r$ whenever $b-(r-1)a$ divides $a$, and show that for $a<b \le 2a$, where this condition reads $b-a=\gcd(a,b)$, it is also necessary. For $b>a(a-1)$, the divisibility condition is superfluous and the Schmidt bound is attained for every such $b$. Thus, for each fixed $a$, attainability of both bounds remains open for only finitely many $b\ge a$.