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

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Pith's one-line read For any n, the eigenvalues of a sum of two n-fold tensor products of positive-semidefinite operators are majorized by the sum of the tensor products of their eigenvalue vectors.

desk verdict Solid extension of Ky Fan majorization to arbitrary tensor factors for two summands, with a clean combinatorial core and an explicit multi-summand counter-example. read the letter →

arxiv 2607.07913 v1 pith:6HVMWNSC submitted 2026-07-08 math.RA math-phmath.COmath.MPquant-ph

Mohammad A. Alhejji , Cole Kelson-Packer This is my paper · ORCID
classification math.RAmath-phmath.COmath.MPquant-ph MSC 15A4215A6947A6390C05
keywords majorizationKyFaninequalitytensorproductspositive-semidefiniteoperatorseigenvalueinequalitieslinearprogrammingquantumentropy
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Ky Fan's classical inequality says that the eigenvalues of a sum of two Hermitian matrices are majorized by the sum of their individual eigenvalue vectors. The authors prove a separable version of that relation: when each summand is itself an n-fold tensor product of positive-semidefinite operators, the same majorization still holds after the tensor products of the eigenvalue vectors are formed. The result immediately yields quantum-entropy inequalities for convex mixtures of density operators that obey tensor-product constraints. A short counter-example shows that the relation fails once three or more such tensor products appear and the number of factors is at least three, so two summands is the natural limit of the statement.

What carries the argument

The linear-programming upper bound υ k(X,Y) on the Ky Fan sums σ k(X+Y), whose feasible region is defined by basic weight constraints and alignment terms that record overlaps of partial eigenspaces; comparing those alignment terms for product bases versus the standard basis yields the desired majorization.

What would settle it

Exhibit positive-semidefinite operators A_i, B_i for which the sum of the k largest eigenvalues of (⊗ A_i)+(⊗ B_i) strictly exceeds the corresponding sum for (⊗ λ(A_i))+(⊗ λ(B_i)), or verify that the alignment terms of the product bases can exceed those of the standard basis.

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Extended reading notes

Core claim

Theorem 1.1 asserts that if A1,…,An and B1,…,Bn are positive-semidefinite operators on spaces of dimensions d1,…,dn, then the eigenvalue vector of (⊗ Ai)+(⊗ Bi) is majorized by (⊗ λ(Ai))+(⊗ λ(Bi)). The relation reduces, via a linear-programming bound on Ky Fan sums, to a majorization between two projectors onto tensor-product bases whose index sets are downward-closed in the product order; that majorization is established by comparing dimensions of subspace intersections and applying a flag-alignment argument.

Load-bearing premise

The reduction rests on a previously established linear-programming characterization of the Ky Fan sums; if that upper bound fails to be tight or even valid for tensor-product operators, the comparison of alignment terms no longer implies the majorization.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 5 minor

Summary. The paper proves Theorem 1.1: for positive integers d1,...,dn and positive-semidefinite operators Ai, Bi on C^{di}, the eigenvalue vector of (⊗_{i=1}^n Ai) + (⊗_{i=1}^n Bi) is majorized by (⊗ λ(Ai)) + (⊗ λ(Bi)). This is obtained by reducing, via the linear-programming upper bounds υ_k on Ky-Fan sums from the authors’ prior work, to a comparison of alignment terms; the comparison is settled by Theorem 1.2 on the eigenvalues of sums of two projectors onto spans of tensor-product basis vectors indexed by downward-closed sets in the product order. The proof of Theorem 1.2 rests on three self-contained combinatorial statements (Proposition 3.2 on cardinalities after local permutations of upward-closed sets, Proposition 3.3 on dimensions of intersections of spans via complete flags, and Lemma 3.4 converting those dimensions into a majorization relation for projectors). Section 4 supplies an explicit numerical counter-example showing that the natural multi-summand extension fails for three or more factors when n≥3.

Significance. The result supplies a clean majorization tool for quantum-entropy inequalities under tensor-product constraints, a setting of current interest. Once the published LP characterization is granted, the remainder of the argument is elementary linear algebra and order theory; the inductive proofs of Propositions 3.2–3.3 and the flag argument are fully detailed and of independent combinatorial interest. The counter-example is concrete and correctly delimits the scope of the claim. These features make the note a solid, usable contribution to majorization theory for multipartite operators.

minor comments (5)
  1. Introduction, first paragraph: the informal allusion to “Robin Hood [5] is elusive” is stylistic and may distract; a more conventional sentence would improve tone for a pure-mathematics audience.
  2. Section 2, after (2.8): the existence of the downward-closed sets Ω_ℓ^{(A)} and Ω_ℓ^{(B)} is asserted “by induction”; a one-sentence sketch of the inductive step (or a reference to the standard fact that the product order is a ranked poset) would make the reduction fully self-contained.
  3. Proposition 3.2, induction step: the successive transposition of the permutation ρ_n is correct but a bit terse; a parenthetical remark that each swap can only decrease (or leave unchanged) the sum would help the reader track the inequality direction.
  4. Example 4.1: the claim that the sum of the three largest eigenvalues exceeds the corresponding sum of coordinates “by at least 0.03” would be more reproducible if the authors indicated the numerical method (exact arithmetic, floating-point precision, or software) used to obtain the bound.
  5. References: several recent arXiv preprints are cited by number only; adding the full titles (already present for some) would aid readers who download the PDF offline.

Circularity Check

1 steps flagged · score 1.0 of 10

No significant circularity: the tensor-product majorization is reduced via a previously published LP bound to a self-contained combinatorial comparison of alignment terms.

  1. self citation load bearing [Sec. 2, reduction of (2.2) to (2.7) via Thm 3.2 of Ref. [2]]
    "We recall the linear programming bounds in Ref. [2]. … The content of Theorem 3.2 in Ref. [2] is that σk(X+Y) is bounded from above by the optimal value of a linear maximization program … To prove (2.2), it suffices to show that υk(⊗Ai,⊗Bi)≤υk(⊗A↓i,⊗B↓i)."

    The comparison of Ky-Fan sums that yields Thm 1.1 is justified solely by the LP characterization proved in the authors’ prior paper. While that characterization is independent of the present tensor-product claim and is therefore legitimate external support, it is still a load-bearing self-citation; if the LP bound failed for tensor-product operators the reduction would collapse. No further circularity is introduced.

full rationale

The paper’s central claim (Thm 1.1) is obtained by reducing the desired majorization to a comparison of Ky-Fan sums σk via the linear-programming upper bound υk(X,Y) established in the authors’ earlier work (Ref. [2], Thm 3.2). Once that characterization is granted, the remainder of the argument is independent: the two linear programs share the same objective, so it suffices to show that every alignment term of the left-hand side is at most the corresponding term of the right-hand side; those inequalities are then proved by Thm 1.2, whose proof relies only on the existence of simultaneous complete flags (Lem 3.1, classical), a purely combinatorial statement about downward- and upward-closed sets under product order (Prop 3.2), a dimension lower bound for intersections of tensor-product subspaces (Prop 3.3), and the elementary majorization lemma for projectors with controlled orthogonal complements (Lem 3.4). No quantity appearing in the final statement is defined in terms of the target majorization, no parameter is fitted, and the only self-citation is the already-published LP tool, which is independent of the tensor-product claim. The multi-summand counter-example is explicit and does not affect the two-summand derivation. Hence the circularity score is at most 1.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper rests entirely on classical linear algebra (complete flags, majorization, Ky Fan) and one previously published LP characterization of Ky-Fan sums. No free parameters or new physical entities are introduced; the only non-standard ingredient is the combinatorial statement about intersections of downward- and upward-closed sets under local permutations, which is proved from first principles.

assumptions (4)
  • standard math Ky Fan’s majorization relation for ordinary sums of Hermitian matrices
    Invoked repeatedly (e.g., Lem 3.4 and the base of the reduction) as a known fact.
  • standard math Existence of a common basis for any two complete flags (Lem 3.1)
    Classical fact used to align the product bases in Prop 3.3; cited to Steinberg and modern accounts.
  • domain assumption Linear-programming upper bound υk(X,Y) for σk(X+Y) (Thm 3.2 of Ref. [2])
    The entire reduction in Sec 2 treats this characterization as given; if it fails, the comparison of alignment terms does not imply the eigenvalue majorization.
  • domain assumption Eigenvalues of a tensor product of PSD operators respect the product order on multi-indices
    Used to guarantee that the largest-eigenvalue subspaces can be chosen downward-closed; elementary but load-bearing for the reduction.

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Pith. "Pith review of A majorization relation for a sum of two tensor products of positive semidefinite operators." pith.science (2026). https://pith.science/paper/6HVMWNSC

@misc{pith2026260707913,
  author       = {Pith},
  title        = {Pith review of: A majorization relation for a sum of two tensor products of positive semidefinite operators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6HVMWNSC}},
  note         = {Machine review of arXiv:2607.07913}
}
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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