Dimension-free marginal inequalities for pure three-party states

The paper in detail

Let \(\psi\in\mathcal H_A\otimes\mathcal H_B\otimes\mathcal H_C\) be a pure state with \(\|\psi\|^2=m\), and let \(a,b,c\) be the decreasingly ordered spectra of its one-party marginals. Which triples \((a,b,c)\) occur? For three qubits the answer is the polygon inequality \(a_2\le b_2+c_2\) and its permutations. For three qutrits Higuchi found seven inequalities up to permutation. In principle Klyachko solved the problem: for fixed dimensions the admissible spectra form a convex polytope whose facets come from Schubert calculus. In practice the lists grow quickly; Bulois, Denis and Ressayre count 47, 462, 5,333 and 64,792 classes of inequalities for \(d=4,5,6,7\).

Few inequalities are known that hold in every dimension. The basic one is Klyachko’s first-row bound \[a_1\le\sum_i\min(b_i,c_i),\qquad\text{hence}\qquad a_1\le m+b_I-c_I\ \text{ for every finite } I.\] The paper first observes that the remaining qutrit inequalities, rewritten as \(a_1-a_2\le m-b_i+c_i+b_j-c_j\) and \(a_1-a_{i+1}\le m-b_{i+j-1}+b_k+c_j-c_k\), hold in all dimensions (Theorem 5).

The main result, Theorem 6, is a lifting rule. Take any valid seed inequality for the smaller state left after removing the largest Schmidt component of party \(A\). Bound the spectra of \(B\) and \(C\) with Horn triples, which control sums of Hermitian matrices independently of dimension. The result is a new inequality for the original state. Lifts can be lifted again. Corollary 1 gives two explicit families, for example \(b_j+c_K\le m+a_{J+1}\) for every Horn triple \((I,J,K)\) and \(j\notin I\). Every inequality obtained this way also holds for partitions \(\lambda,\mu,\nu\vdash n\) with nonzero Kronecker coefficient, with \(m\) replaced by \(n\).

For local dimension 4, all 50 facet classes of the marginal polytope follow: 11 are trivial or first-row bounds, 32 need one lifting step and 7 need two. In dimension 5, 455 of 466 classes follow from at most two steps. The other 11 are not implied at any depth, which the author shows by exhibiting points that satisfy every lifted inequality but lie outside the polytope. Since the family is computable in polynomial time, it gives a usable outer approximation where exact facet lists are out of reach.

Transferable methods

1. Peel off the largest Schmidt component

   The recursion removes one Schmidt term of party \(A\) and tracks how the marginals of \(B\) and \(C\) change. It turns a three-party question into one about sums of positive matrices.

   Write the Schmidt decomposition across \(A|BC\) as \(\psi=\sum_j\sqrt{a_j}\,e_j\otimes\varphi_j\). Because the \(e_j\) are orthonormal, the cross terms drop out of the partial trace over \(A\), and \[\rho_B=\underbrace{a_1\operatorname{Tr}_C\ketbra{\varphi_1}{\varphi_1}}_{P}+\rho_B^{\prime},\qquad\rho_C=\underbrace{a_1\operatorname{Tr}_B\ketbra{\varphi_1}{\varphi_1}}_{P^{\prime}}+\rho_C^{\prime},\] where \(\rho^{\prime}\) are the marginals of the child \(\psi^{\prime}=\sum_{j\ge2}\sqrt{a_j}\,e_j\otimes\varphi_j\). The child has mass \(m-a_1\) and \(A\)-spectrum \(\hat a=(a_2,a_3,\dots)\). \(P\) and \(P^{\prime}\) are the two marginals of the bipartite pure vector \(\sqrt{a_1}\varphi_1\), so they share one spectrum \(\tau\) with \(\sum_i\tau_i=a_1\) (Lemma 3).

   Klyachko’s bound follows in one line (editorial presentation of the paper’s argument). Since \(\rho_B\ge P\) and \(\rho_C\ge P^{\prime}\), Weyl monotonicity gives \(b_i\ge\tau_i\) and \(c_i\ge\tau_i\). So \(\min(b_i,c_i)\ge\tau_i\), and summing gives \(\sum_i\min(b_i,c_i)\ge a_1\). We checked this, and the lifted inequality \(b_2+c_1\le1+a_2\), on random states of local dimension 2 to 4.

   How to reuse it

   Use this whenever a constraint on several marginals is wanted. Removing one Schmidt term writes each other marginal as a known-spectrum piece plus the marginal of a smaller instance. Horn’s problem, which is completely solved, then controls the sum, and induction on the Schmidt rank carries the bound to all dimensions.

2. Combine Horn triples with Abel summation

   The lifting theorem needs to absorb the unknown spectrum \(\tau\) of the peeled piece. It does so with one summation-by-parts bound.

   For \(C=A+B\) with spectra \(c,a,b\), a Horn triple \((I,J,K)\) gives \(c_K\le a_I+b_J\), independently of dimension. The simplest case is Weyl’s inequality \(c_{i+j-1}\le a_i+b_j\). Applied to \(b=\operatorname{spec}(P+\rho_B^{\prime})\), it gives \(b_K\le\tau_I+\beta_J\); monotonicity gives \(b_L\ge\tau_L\). After summing, the \(B\) and \(C\) terms of the target are at most \(\langle\delta,\tau\rangle+\langle u,\beta\rangle+\langle v,\gamma\rangle\), where \(\delta\) collects the \(\tau\)-indices that were used.

   Because \(\tau\) is decreasing and nonnegative, summation by parts gives \[\langle\delta,\tau\rangle=\sum_{k\ge1}(\tau_k-\tau_{k+1})\sum_{j=1}^{k}\delta_j\ \le\ \bar\kappa\sum_{k\ge1}k(\tau_k-\tau_{k+1})=\bar\kappa\,|\tau|=\bar\kappa\,a_1,\] provided every prefix sum satisfies \(\sum_{j\le k}\delta_j\le\bar\kappa k\). The seed applied to the child bounds the remaining terms by \(\kappa(m-a_1)+\langle w,\hat a\rangle\). Adding the pieces gives the lifted inequality (29).

   How to reuse it

   Abel summation is a general way to bound a linear form over an unknown decreasing vector by its total: it is enough to check the prefix sums of the coefficients. It applies to eigenvalue and majorization problems in which one spectrum is known only through its trace.

Source: Sections II–V: Lemmas 1–3, Theorems 5–6 and Corollary 1; Section VI. Based on the linked paper version. Derivations marked as editorial, and suggested uses in other problems, are editorial guidance rather than results of the paper.

Context & limitations

Why it matters. Gives efficiently computable, dimension-independent outer approximations to the set of compatible marginals, whose exact facets are known only in small dimensions.

Limitations. Eleven facet classes in dimension 5 are not implied, so the family is incomplete for every dimension of 5 or more. The comparisons in dimensions 4 and 5 are computer-assisted.

Description based on the full text of the linked arXiv version, read 2026-10-21; equations follow the paper’s notation.