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

Primarily human-written textHuman understanding: all partsmath.OA — Operator Algebrasmath.FA — Functional Analysisquant-ph — Quantum Physics

Contributed by Dipankar Dey, Somnath Jana, Soumyashant Nayak ↗

Version 1 / Oct 23, 2026 / CC BY 4.0

Abstract

Let 2≤a≤b2 \le a \le b, and let ρA,ρB\rho_A, \rho_B be density operators on $\C^a$ and $\C^b$, respectively. Let C(ρA,ρB)\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)(a, b) this bound is tight. We restrict attention to the tracial case ρA=Ia/a\rho_A = I_a/a, ρB=Ib/b\rho_B = I_b/b. For the Parthasarathy bound, we construct extreme couplings of rank rr from pairs of length-rr finite unit-norm tight frames (FUNTFs) in $\C^a$ and $\C^b$ satisfying a transversality condition. This yields an extreme coupling of rank bb for every 2≤a<b2 \le a < b, and hence tightness of $\KRP(a,b)$ whenever a<ba<b and b≥(a2−1)/2b\ge(a^2-1)/2; in particular, along the whole qubit-qudit line (2,b)(2,b), b≥3b\ge3. This recovers the two previously known tracial cases with a<ba<b, namely (2,3)(2,3) and (3,4)(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 rr whenever b−(r−1)ab-(r-1)a divides aa, and show that for a<b≤2aa<b \le 2a, where this condition reads b−a=gcd⁡(a,b)b-a=\gcd(a,b), it is also necessary. For b>a(a−1)b>a(a-1), the divisibility condition is superfluous and the Schmidt bound is attained for every such bb. Thus, for each fixed aa, attainability of both bounds remains open for only finitely many b≥ab\ge a.

Provenance statement

The main mathematical ideas, constructions, and proof strategy in this work are due to the authors. Moonshot AI's Kimi K3 and Google's Gemini 3.1 Pro were used in organizing and restructuring parts of the proofs, including separating intermediate arguments into lemmas, and in generating examples. The authors critically assessed and developed these suggestions and, in several cases, substantially refined the arguments. The authors take full responsibility for the correctness of the mathematical contents of the paper.

Tools used

Moonshot AI
KimiVersion K3
Google
GeminiVersion 3.1 Pro

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Version history

  1. v1Submitted by Soumyashant NayakInitial depositCurrentOct 23, 2026