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arXiv:2607.07913

A majorization relation for a sum of two tensor products of positive semidefinite operators

Mohammad A. Alhejji, Cole Kelson-Packer

math.RAmath-phmath.COquant-phSubmitted Jul 8, 2026• Announced Jul 10, 2026
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Abstract

We use linear programming to prove a separable version of Ky Fan's majorization relation for a sum of two operators that are each a tensor product of $n$ positive semidefinite operators. We give an example showing that such a relation does not hold in general for sums of three or more tensor products of three or more positive semidefinite operators.

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10 pages, no figures

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We use linear programming to prove a separable version of Ky Fan's majorization relation for a sum of two operators that are each a tensor product of nnn positive semidefinite operators. We give an example showing that such a relation does not hold in general for sums of three or more tensor products of three or more positive semidefinite operators.
Several classical majorization relations for sums or products of matrices involve a majorizing vector of perfectly aligned eigenvalues or singular values. By relaxing the order of alignment to a partial order, we show that the majorization can be strengthened, provided the change-of-basis matrices admit an LU-approximation with respect to this partial order. In this way, we obtain refined versions of Ky Fan's majorization relations, Horn's log-majorization relation, and von Neumann's trace inequality. As an application, we give a short proof of the separable Ky Fan majorization relation for an arbitrary number of tensor factors and extend it to a sum of tensor products of arbitrary matrices. Further applications concern majorization relations for sums of (anti-)symmetric powers and for products of Kronecker sums.
Abstract A separable version of Ky Fan’s majorization relation is proven for a sum of two operators that are each a tensor product of two positive semi-definite operators. In order to prove it, upper bounds are established on the relevant largest eigenvalue sums in terms of the optimal values of certain linear programs. The objective function of these linear programs is the dual of the direct sum of the spectra of the summands. The feasible sets are bounded polyhedra determined by positive numbers, called alignment terms, that quantify the overlaps between pairs of largest eigenvalue spaces of the summands. By appealing to geometric considerations, tight upper bounds are established on the alignment terms of tensor products of positive semi-definite operators. As an application, the spin alignment conjecture in quantum information theory is affirmatively resolved to the 2-letter level. Consequently, the coherent information of platypus channels is additive to the 2-letter level.
Let K, K' be convex cones residing in finite-dimensional real vector spaces E, E'. An element in the tensor product E \otimes E' is K \otimes K'-separable if it can be represented as finite sum \sum_l x_l \otimes x'_l with x_l \in K and x_l' \in K' for all l. Let S(n), H(n), Q(n) be the spaces of n x n real symmetric, complex hermitian and quaternionic hermitian matrices, respectively. Let further S_+(n), H_+(n), Q_+(n) be the cones of positive semidefinite matrices in these spaces. If a matrix in H(mn) = H(m) \otimes H(n) is H_+(m) \otimes H_+(n)-separable, then it fulfills also the so-called PPT condition, i.e. it is positive semidefinite and has a positive semidefinite partial transpose. The same implication holds for matrices in the spaces S(m) \otimes S(n), H(m) \otimes S(n), and for m < 3 in the space Q(m) \otimes S(n). We provide a complete enumeration of all pairs (n,m) when the inverse implication is also true for each of the above spaces, i.e. the PPT condition is sufficient for separability. We also show that a matrix in Q(n) \otimes S(2) is Q_+(n) \otimes S_+(2)-separable if and only if it is positive semidefinite.
For a positive integer nnn, let MnM_nMn​ be the set of n×nn\times nn×n complex matrices. Suppose ∥⋅∥\|\cdot\|∥⋅∥ is the Ky Fan kkk-norm with 1≤k≤mn1 \le k \le mn1≤k≤mn or the Schatten ppp-norm with 1≤p≤∞1 \le p \le \infty1≤p≤∞ (p≠2p\ne 2p=2) on MmnM_{mn}Mmn​, where m,n≥2m,n\ge 2m,n≥2 are positive integers. It is shown that a linear map ϕ:Mmn→Mmn\phi: M_{mn} \rightarrow M_{mn}ϕ:Mmn​→Mmn​ satisfying ∥A⊗B∥=∥ϕ(A⊗B)∥forallA∈MmandB∈Mn\|A\otimes B\| = \|\phi(A\otimes B)\| for all A \in M_m and B \in M_n∥A⊗B∥=∥ϕ(A⊗B)∥forallA∈Mm​andB∈Mn​ if and only if there are unitary U,V∈MmnU, V \in M_{mn}U,V∈Mmn​ such that ϕ\phiϕ has the form A⊗B↦U(φ1(A)⊗φ2(B))VA\otimes B \mapsto U(\varphi_1(A) \otimes \varphi_2(B))VA⊗B↦U(φ1​(A)⊗φ2​(B))V, where φs(X)\varphi_s(X)φs​(X) is either the identity map X↦XX \mapsto XX↦X or the transposition map X↦XtX \mapsto X^tX↦Xt. The results are extended to tensor space Mn1⊗⋯⊗MnmM_{n_1} \otimes \cdots \otimes M_{n_m}Mn1​​⊗⋯⊗Mnm​​ of higher level. The connection of the problem to quantum information science is mentioned.
Tensor product operators on finite dimensional Hilbert spaces are studied. The focus is on bilinear tensor product operators. A tensor product operator on a pair of Hilbert spaces is a maximally general bilinear operator into a target Hilbert space. By 'maximally general' is meant every bilinear operator from the same pair of spaces to any Hilbert space factors into the composition of the tensor product operator with a uniquely determined linear mapping on the target space. There are multiple distinct tensor product operators of the same type; there is no "the" tensor product. Distinctly different tensor product operators can be associated with different parts of a multipartite system without difficulty. Separability of states, and locality of operators and observables is tensor product operator dependent. The same state in the target state space can be inseparable with respect to one tensor product operator but separable with respect to another, and no tensor product operator is distinguished relative to the others; the unitary operator used to construct a Bell state from a pair of |0>'s being highly tensor product operator-dependent is a prime example. The relationship between two tensor product operators of the same type is given by composition with a unitary operator. There is an equivalence between change of tensor product operator and change of basis in the target space. Among the gains from change of tensor product operator is the localization of some nonlocal operators as well as separability of inseparable states. Examples are given.