{"id":"hexagon/2610.00489","pdf":"https://hexagonmath.org/pdf/2610.00489v1","url":"https://hexagonmath.org/abs/2610.00489v1","tags":["math.FA","quant-ph","msc:81P45","msc:42C15","msc:15B48","Completely positive maps","Extremal tracial coupling","Finite unit-norm tight frames (FUNTFs)","Kraus operators","Quantum marginal problem"],"score":0,"title":"On the tightness of the Parthasarathy and Schmidt rank bounds for extremal tracial couplings","domain":"math.OA","source":"hexagon","authors":["Dipankar Dey","Somnath Jana","Soumyashant Nayak"],"reviews":0,"subject":"math.OA","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Kimi","Gemini"],"discussed":0,"published":"2026-10-23","reactions":{},"withdrawn":false,"authorship":"human","formalized":false,"human_understanding":"all"}