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Dimension-free marginal inequalities for pure three-party states

This paper introduces a novel, dimension-independent family of inequalities that bound the local spectra of pure tripartite quantum states, providing a complete set of eigenvalue constraints for systems with local dimensions up to four.

Original authors: Simon Morelli

Published 2026-10-21

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CC BY 4.0

Original authors: Simon Morelli

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

In the quantum world, the whole is often more than the sum of its parts, but it is also strictly bound by them. Imagine a complex system made of three distinct pieces, like a trio of particles linked together in a single, pure state. While we can measure the properties of each individual piece, those local measurements are not free to be anything we like; they must fit together to form a coherent global picture. This is the heart of the quantum marginal problem, a fundamental question in physics asking which local properties are compatible with a specific global state. It is a question of great importance because in many real-world experiments, scientists can only access parts of a complex system, yet they need to understand the whole. If the local pieces do not fit the rules of the global whole, the state cannot exist.

For decades, physicists have known that for small systems, there are strict rules governing how these local properties must align. However, as the systems grow larger, the number of rules explodes, and calculating them becomes nearly impossible. The mathematical landscape is filled with a vast, shifting polytope of possibilities, and for a long time, no one had a way to map its boundaries for systems of arbitrary size. Researchers could list the rules for tiny systems, but they lacked a general method to describe the limits for larger, more complex ones. This gap left a significant blind spot in our understanding of how quantum correlations work across different scales.

A new study by Simon Morelli at the Vienna Center for Quantum Science and Technology has introduced a powerful new family of rules that can be applied to quantum systems of any size. The researcher developed a method to generate a set of inequalities—mathematical limits that define what is possible and what is forbidden—for the local properties of a three-party quantum system. Unlike previous approaches that required re-calculating everything from scratch for every new system size, these new rules work for systems of any dimension. They provide a constructive way to build a map of the allowed states, offering a reliable outer boundary that tells us which combinations of local properties can exist together in a pure quantum state.

The core of this discovery lies in a clever recursive strategy. The researcher realized that one could peel away the most significant part of one particle's state and examine what remains. By treating the remaining part as a smaller, "child" system, the rules for the larger system can be built up from the rules of the smaller one. This process connects the quantum marginal problem to a different, well-understood mathematical area concerning how matrices add together. By leveraging this connection, the new method generates a sequence of inequalities that hold true regardless of how large the system becomes. These inequalities act as a filter, catching any combination of local properties that would violate the laws of quantum mechanics, even if the full, complete list of rules for that specific size is too vast to compute.

The results are particularly striking for systems of small to medium size. For systems where each party has a dimension of four or less, this new family of rules is complete; it generates every single necessary condition, leaving no gaps in the description of the allowed states. This means that for these sizes, the researcher has effectively solved the problem of characterizing the possible local spectra. However, as the systems grow larger, specifically reaching a dimension of five, the method begins to miss a small fraction of the necessary rules. While the new inequalities still capture the vast majority of constraints and provide a very tight approximation of the allowed states, they do not yet form the complete, exhaustive list for these larger dimensions.

Despite this limitation in higher dimensions, the work represents a decisive step forward. The new inequalities can be computed efficiently, making them a practical tool for physicists who need to check if a proposed quantum state is valid without getting bogged down in impossible calculations. By providing a dimension-free framework, the study offers a clear, unified view of how local and global properties interact in pure quantum systems. It transforms a problem that was previously intractable for large systems into one where we can at least see the boundaries of possibility, bringing us closer to a full understanding of the intricate structure of quantum correlations.

Technical Summary: Dimension-free marginal inequalities for pure three-party states

Problem Statement
The paper addresses the Quantum Marginal Problem (QMP) for pure tripartite quantum states. The QMP seeks to determine the compatibility between the local reduced density matrices (marginals) of a multipartite system and the global pure state. Specifically, for a pure state $|\psi\rangle \in \mathbb{C}^{d_A} \otimes \mathbb{C}^{d_B} \otimes \mathbb{C}^{d_C}$ , the problem asks which triples of local spectra $(a, b, c)$ are "admissible." While the set of admissible spectra forms a convex polytope (the marginal polytope) for fixed dimensions, a complete characterization via linear inequalities is known only for small dimensions (e.g., $d \leq 3$ ). For higher dimensions, calculating the facets of this polytope becomes computationally infeasible due to the exponential growth of inequalities derived from Schubert calculus and Kronecker coefficients. Furthermore, existing results are often dimension-specific, lacking a unified, dimension-free family of inequalities.

Methodology and Algebraic Framework
The paper utilizes the algebraic framework connecting the QMP to representation theory. It distinguishes between two related eigenvalue problems:

Horn's Problem: Concerns the spectra of $A+B$ , governed by Littlewood–Richardson coefficients ( $c^\nu_{\lambda\mu}$ ).
Tripartite Pure QMP: Concerns the spectra of marginals of a pure state, governed by Kronecker coefficients ( $g(\lambda, \mu, \nu)$ ).

The paper leverages the correspondence between admissible spectra and the support of Kronecker coefficients (Theorem 2). A key tool is Dvir's recursion, which relates Kronecker coefficients to Littlewood–Richardson coefficients. The author adapts a recursive technique known as Siewert's recursion (Lemma 3), which involves "peeling" the largest eigenvalue $a_1$ from the spectrum of party $A$ . This process decomposes the original state into a "child" state with reduced mass ( $m - a_1$ ) and modified spectra, allowing the application of known inequalities to the child state to derive constraints on the original state.

Key Contributions
The primary contribution is the introduction of a novel, constructive family of dimension-free inequalities bounding the allowed local spectra of pure tripartite states.

General Lifting Theorem (Theorem 6): The author establishes a "lifting" mechanism. Starting from a "seed" inequality valid for a child state (e.g., a bound on the first row of the spectrum), they derive new inequalities for the parent state. This is achieved by:
- Peeling the largest eigenvalue $a_1$ .
- Bounding the remaining terms using Horn inequalities (which are dimension-independent).
- Applying the seed inequality to the child state.
- Combining these to form a linear inequality involving the original spectra $(a, b, c)$ and the total mass $m$ .
New Families of Inequalities:
- The paper generalizes known inequalities for $d=3$ (Higuchi's inequalities) to arbitrary dimensions.
- It introduces new inequalities (e.g., Eq. 20 and Eq. 21) that were previously unknown for general dimensions.
- Corollary 1 provides explicit forms of these lifted inequalities, showing how Horn triples $(I, J, K)$ can be used to generate bounds on the spectra.
Dimension-Free Validity: Unlike previous characterizations that require specific dimension constraints, these inequalities hold for arbitrary finite local dimensions.

Results and Completeness
The author analyzes the completeness of their derived inequalities by comparing them against known facet lists for specific dimensions:

Dimensions 2 and 3: The new family of inequalities generates the complete set of admissible spectral constraints (reproducing the polygon inequality for qubits and Higuchi's 7 inequalities for qutrits).
Dimension 4: Through computer-assisted exact calculations, the author verifies that their inequalities generate all 50 classes of facets for the marginal polytope.
Dimension 5 and Higher: The family is incomplete. The author identifies 11 classes of inequalities in dimension 5 that are not implied by their recursive lifting method. They demonstrate this by finding points outside the quantum marginal polytope that satisfy all inequalities derived from their method but are excluded by the known facets.

Significance and Claims
The paper claims that this work establishes a fundamental connection between Horn's problem and the pure tripartite QMP. The significance lies in:

Constructive Generation: Providing a systematic, polynomial-time computable method to generate outer approximations of the set of allowed marginals for any dimension.
Partial Characterization: While not exhaustive for $d \geq 5$ , the inequalities fully describe the marginal polytope for $d \leq 4$ and recover a large fraction of the constraints for higher dimensions.
Theoretical Insight: The work clarifies the structural relationship between the asymptotic support of Kronecker coefficients and linear spectral inequalities, offering a new perspective on the "peeling" of eigenvalues in multipartite systems.

The author maintains a modest stance, acknowledging that while their inequalities do not provide a complete characterization for all dimensions, they offer a powerful, dimension-independent tool for approximating the quantum marginal polytope, which is practically relevant for systems where only local reduced density matrices are accessible.

Technical Summary: Dimension-free marginal inequalities for pure three-party states

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