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

This paper utilizes linear programming to establish a separable version of Ky Fan's majorization relation for the sum of two tensor products of positive semidefinite operators, while demonstrating that such a relation fails for sums involving three or more tensor products.

Original authors: Mohammad A. Alhejji, Cole Kelson-Packer

Published 2026-07-10

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Mohammad A. Alhejji, Cole Kelson-Packer

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a massive, multi-layered Lego castle. Each layer is built from a different set of bricks, and the whole structure is a "tensor product"—a fancy way of saying the layers are stacked together so tightly that they act as one giant, complex object. Now, imagine you have two of these castles, let's call them Castle A and Castle B. Both are made of "positive semidefinite" bricks, which is a math-speak way of saying they are solid, stable, and never have negative weight.

The big question the authors, Mohammad Alhejji and Cole Kelson-Packer, asked was: If you smash Castle A and Castle B together into a single pile, can you predict the "shape" of the new pile just by looking at the shapes of the original two?

The Big Discovery: A Perfect Match for Two Castles
The authors proved that for exactly two castles, the answer is a resounding "yes." They showed that the "heaviness" (or eigenvalues, which are like the brightness levels of the castle's lights) of the combined pile is always "majorized" by a specific, simpler version of the sum.

Think of "majorization" like a game of musical chairs with energy. If you have a pile of energy from Castle A and a pile from Castle B, the authors proved that when you mix them, the resulting energy distribution is always "flatter" or "more spread out" than if you had just lined up the brightest lights from A and the brightest lights from B side-by-side.

They didn't just guess this; they used a powerful tool called linear programming (a method for finding the best outcome in a mathematical model) to prove it. They broke the problem down into smaller pieces, looking at how the "downward-closed" sets of bricks (imagine a pyramid where if you have a brick, you must have all the bricks below it) overlap. They showed that no matter how you arrange the bricks in the two original castles, the combined pile's energy profile will always be "less extreme" than the theoretical maximum you'd get if you just added the sorted lists of their brightest spots.

The "Nope" List: Why Three is a Crowd
Here is where the story gets tricky. The authors were very careful to tell us that this beautiful rule does not work if you try to smash three or more castles together.

They explicitly ruled out the idea that this pattern continues forever. If you take three tensor products (three different multi-layered castles) and add them up, the neat majorization relation breaks down. To prove this, they didn't just say "it probably doesn't work"; they built a specific counterexample.

They constructed a scenario with three specific quantum states (represented by complex numbers and vectors like $|e_1\rangle$ and $|e_2\rangle$ ) in a 2-dimensional space. When they added up the three largest eigenvalues of the combined sum, the number was at least 0.03 higher than the sum of the three largest coordinates of the individual parts added up separately. This tiny gap of 0.03 is the smoking gun that proves the rule fails for three or more items. It's like trying to predict the weather by adding up three different forecasts and finding out the actual storm is slightly stronger than the sum of the predictions.

How Sure Are They?
The authors are extremely confident about the two-item case. They didn't just simulate it; they provided a rigorous mathematical proof using linear programming and properties of eigenbases. They stated clearly that their method works for any number of layers ( $n$ ) as long as you are only adding two operators.

However, for the case of three or more operators, they are equally confident that the rule fails, but for a different reason: they provided a concrete counterexample. They didn't suggest it might fail; they showed a specific instance where it definitely does.

The Takeaway
So, if you have two quantum "castles" made of tensor products, you can safely predict their combined behavior using this majorization rule. It's a reliable map for a two-person dance. But if you invite a third dancer, the choreography changes, and the map no longer works. The authors have drawn a hard line in the sand: this relation holds for two, but it does not generalize to three or more. They even noted that the case of two layers ( $n=2$ ) with three or more summands ( $m \ge 3$ ) is still an open question, meaning we don't know the answer for that specific mix yet.

In short: Two is a pair, and the math holds up. Three is a crowd, and the math breaks.

Technical Summary: A Majorization Relation for a Sum of Two Tensor Products of Positive Semidefinite Operators

Problem Statement
The paper addresses the problem of establishing majorization relations for sums of tensor products of positive semidefinite (PSD) operators. Specifically, it investigates whether the eigenvalue vector of a sum of two tensor products, $\bigotimes_{i=1}^n A_i + \bigotimes_{i=1}^n B_i$ , is majorized by the tensor product of the sums of their individual eigenvalue vectors, $\bigotimes_{i=1}^n \lambda(A_i) + \bigotimes_{i=1}^n \lambda(B_i)$ .

While the case for $n=1$ is a known restatement of Ky Fan's majorization relation, and the case for $n=2$ was previously addressed using linear programming (Ref. [2]), the extension to general $n$ remained unresolved. The authors note that the proof technique used for $n=2$ in Ref. [2] fails to generalize to $n > 2$ , specifically regarding the proof of Proposition 4.2 in that reference. Furthermore, the paper questions whether such a relation holds for sums of three or more tensor products or for tensor products involving three or more factors in a sum of three or more terms.

Methodology
The authors employ a linear programming approach combined with combinatorial arguments on partially ordered sets. The methodology proceeds in three main stages:

Reduction to Projectors: The authors reduce the general eigenvalue inequality (Theorem 1.1) to a problem involving sums of orthogonal projectors. By utilizing the linearity of the trace and the properties of Ky Fan's norms ( $\sigma_k$ ), they show that proving the majorization relation is equivalent to proving that the optimal value of a specific linear program for the sum of tensor products is bounded by the optimal value for a "diagonalized" version. This reduces the problem to comparing alignment terms involving projectors onto subspaces spanned by eigenvectors corresponding to the largest eigenvalues.
Combinatorial Analysis of Downward-Closed Sets: The core of the proof relies on the structure of tensor-product bases. The authors define sets of indices $\Omega(A)$ and $\Omega(B)$ corresponding to the eigenspaces of the largest eigenvalues. Due to the product order of eigenvalues in tensor products of PSD operators, these index sets are "downward-closed" in the Cartesian product of index sets $\Pi_{i=1}^n [d_i]$ .
Intersection Bounds via Linear Algebra and Permutations: To establish the majorization, the authors prove a key geometric lemma (Theorem 1.2) concerning the intersection of subspaces spanned by two different tensor-product bases restricted to downward-closed sets.
- They utilize a result regarding complete flags (Lemma 3.1) to align two arbitrary tensor-product bases with a common reference basis via permutations.
- They prove a combinatorial proposition (Proposition 3.2) stating that the intersection size of a downward-closed set and a locally-permuted upward-closed set is bounded below by the intersection of the original sets.
- Combining these, they derive a lower bound on the dimension of the intersection of the support of one projector and the kernel of another (Proposition 3.3).
- Finally, they apply a lemma (Lemma 3.4) relating subspace intersection dimensions to the eigenvalue distribution of the sum of two projectors, showing that the eigenvalues of the sum of the arbitrary projectors are majorized by the eigenvalues of the sum of the "standard" projectors (which correspond to the right-hand side of the inequality).

Key Contributions and Results

Theorem 1.1: The primary result is the proof that for any positive integers $d_1, \dots, d_n$ and PSD operators $A_i, B_i$ acting on $\mathbb{C}^{d_i}$ , the following majorization holds:
$\lambda\left( \bigotimes_{i=1}^n A_i + \bigotimes_{i=1}^n B_i \right) \preceq \bigotimes_{i=1}^n \lambda(A_i) + \bigotimes_{i=1}^n \lambda(B_i)$
This implies corresponding inequalities for unitarily invariant concave functions, such as quantum entropy, for convex mixtures of density operators subject to tensor-product constraints.
Theorem 1.2: A more general technical result is established regarding the majorization of sums of projectors defined on downward-closed sets in tensor-product spaces. This theorem serves as the engine for the main result and generalizes the linear programming bounds used in the $n=2$ case.
Counterexample for Generalization: The paper demonstrates that the relation does not generalize to sums of three or more tensor products when $n \ge 3$ . Example 4.1 provides a specific counterexample in $\mathbb{C}^2$ involving three summands, showing that the sum of the 3 largest eigenvalues of the operator sum exceeds the sum of the 3 largest coordinates of the corresponding eigenvalue sum by at least 0.03.

Significance and Claims
The authors frame the discovery as a significant addition to the relatively rare class of majorization relations, which are powerful tools for proving inequalities in quantum information theory. The work specifically resolves the gap left by previous literature regarding the $n > 2$ case for sums of two tensor products.

The paper claims that this result implies quantum-entropy inequalities for convex mixtures of density operators under tensor-product constraints. However, the authors remain modest regarding the scope of the result, explicitly stating that the relation fails for sums of three or more terms and noting that the case of $n=2$ with $m \ge 3$ summands remains an open question. The work does not propose new experimental setups or applications beyond the theoretical implications for entropy inequalities and the mathematical structure of operator majorization.

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