Moment Polytopes of Tensors

• Moment polytopes of tensors are convex polytopes that capture quantum marginal spectra and invariant-theoretic data from multilinear group actions.
• Explicit geometric constructions and convex optimization techniques enable efficient computation of these polytopes for complex tensor formats such as 3×3×3 and 4×4×4.
• Applications in quantum information, algebraic complexity, and tensor networks illustrate their role in linking entanglement, tensor decompositions, and computational complexity.

Moment polytopes of tensors encode the asymptotic spectra of quantum marginals and invariant-theoretic data arising from multilinear actions of reductive groups on tensor spaces. Such polytopes have become central objects in representation theory, algebraic complexity, and quantum information theory, unifying geometric, combinatorial, and computational perspectives. Their paper encompasses explicit geometric constructions, convex optimization, combinatorial descriptions (orbit and support-based), and deep links to tensor decompositions, entanglement, and asymptotic rank theory. Recent advances illuminate the structure, computation, and applications of moment polytopes of tensors, especially for spaces such as $$\mathbb{C}^n \otimes \mathbb{C}^n \otimes \mathbb{C}^n$$.

1. Foundational Principles: Definitions and Constructions

The moment polytope $\Delta(T)$ associated to a tensor $$T \in \mathbb{C}^a \otimes \mathbb{C}^b \otimes \mathbb{C}^c$$ is a convex polytope defined via the spectra of its normalized marginals under local actions of $GL_a \times GL_b \times GL_c$ (or their subgroups). Formally, for each local mode $i = 1,2,3$, one computes the "quantum marginal" $\mu_i(T)$ as the normalized spectrum of contraction along the other modes: $$\mu_i(T) = \frac{T_i^* T_i}{\operatorname{Tr}(T_i^* T_i)}$$ where $T_i$ is the linear map induced by $T$ along index $i$. The moment polytope is then

$$\Delta(T) = \{\operatorname{spec}(\mu(S)) : S \in \overline{GL \cdot T}\setminus \{0\} \}$$

with the marginal spectra projected to the positive Weyl chamber $\mathcal{W}$.

For tensor varieties $$W \subset \mathbb{C}^{a} \otimes \mathbb{C}^{b} \otimes \mathbb{C}^{c}$$ invariant under $GL_a \times GL_b \times GL_c$, the moment polytope is similarly defined as the set of triple marginal spectra attained by tensors in $W$.

Representation-theoretic formalisms further express the polytope as the closure of all normalized highest weights appearing with nonzero multiplicity in tensor powers of $T$: $\Delta(T) = \overline{\{ (\lambda/n, \mu/n, \nu/n) : \text{the %%%%14%%%%-isotypic projection onto %%%%15%%%% is nonzero} \}}$

2. Tensor Products, Minkowski Sums, and Spectral Combinatorics

Moment polytopes feature rich additive properties under tensor operations. For representations $T_1$, $T_2$ of a reductive group $G$, the polytope associated to the tensor product is the Minkowski sum of individual polytopes, as formalized in

$A_w(T_1 \otimes T_2) = A_w(T_1) + A_w(T_2)$

where $A_w(T)$ denotes the weight polytope of $T$. This additivity transfers directly to the moment polytope via intersection with the Weyl chamber $C$: $$A_{\text{mom}}(T_1 \otimes T_2) = (A_w(T_1) + A_w(T_2)) \cap C$$ This principle generalizes to more complex tensorial settings: for Mirković–Vilonen (MV) polytopes associated to crystals, the MV polytope of a tensor product lies inside the Minkowski sum of the polytopes of its factors ($P \subseteq P_1 + P_2$) (Kato et al., 2010). This result exposes the deep intertwining of crystal combinatorics, convex geometry, and tensor representation theory.

3. Computational Methods and Explicit Algorithms

Efficient computation of moment polytopes of tensors, especially in formats beyond $2 \times 2 \times 2$, has posed significant challenges due to exponential growth in facets and orbit complexity. The algorithm of (Berg et al., 9 Oct 2025) leverages Franz's support description and a separation of "attainable inequalities" to compute polytopes for tensors of dimension an order of magnitude larger than previous methods.

This algorithm proceeds in two main phases:

• Combinatorial Enumeration: Candidate facet inequalities are generated as solutions to linear systems involving affine constraints and symmetries of the tensor's support. This reduces the enumeration from an exponential ($2^{abc}$) to a manageable polynomial scale via symmetry reductions.
• Attainability Verification: Each candidate inequality is tested for attainability using polynomial equations representing vanishing conditions for tensor entries in certain half-spaces, decided via Gröbner basis computations over $\mathbb{Q}$ or finite fields. The set of attainable inequalities defines a candidate polytope, which is verified by tensor scaling and convex optimization to ensure all its vertices are genuinely marginal spectra of orbit-closure representatives.

This framework enables, for the first time, explicit computation of all moment polytopes for tensors in $3 \times 3 \times 3$ and, with high probability, in $4 \times 4 \times 4$—including the $2 \times 2$ matrix multiplication tensor.

4. Applications: Quantum Marginals, Complexity, and Tensor Networks

Moment polytopes serve as geometric invariants in quantum information, algebraic complexity, and optimization:

• Quantum Marginals/Entanglement Polytopes: They provide a solution to the quantum marginal problem for single-particle reduced densities of multipartite pure states, classifying which joint spectra are attainable.
• Algebraic Complexity Theory: In the context of geometric complexity theory and matrix multiplication, moment polytopes characterize quantum functionals that serve as obstructions in asymptotic spectrum theory, particularly for separating tensor complexity classes (Berg et al., 28 Mar 2025).
• Tensor Networks: Results show that tensor networks with genuine multiparty entanglement (GHZ-type unit tensors) achieve strictly larger moment polytopes than networks built from matrix multiplication tensors (PEPS), revealing greater expressivity (Berg et al., 28 Mar 2025).

Tables below illustrate protocol classes and polytope relationships for low-dimensional tensor formats.

Tensor Type	Moment Polytope Maximality	Attainable Marginals
Matrix multiplication ($M_n$)	Not maximal (Berg et al., 28 Mar 2025)	Proper subset of Kronecker
Unit tensor ($U(r)$)	Maximal (small $r$)	Contains uniform marginals
MV polytope (crystal)	Minkowski Additive (Kato et al., 2010)	Constrained by GGMS data

A central numerical and geometric consequence is that certain uniform marginal spectra, obtainable by the unit tensor, cannot be realized by matrix multiplication tensors for moderate ranks, signifying strict separation in attainable quantum states and computational complexity (Berg et al., 28 Mar 2025).

5. Operational and Decision-Theoretic Aspects

The problem of deciding membership in a moment polytope (given a candidate spectrum) is rigorously analyzed in (Bürgisser et al., 2015). The main result is that, for representations of a compact, connected Lie group, this decision problem is in NP and coNP. This is established via dual certificates:

• NP certificate: An explicit tensor realization with prescribed marginals.
• coNP certificate: A nontrivial separating facet defined by a Ressayre element.

This is particularly striking given the NP-hardness of deciding the positivity of individual Kronecker coefficients; by contrast, inclusions of stretched Kronecker polytopes are fully tractable, providing deep computational implications for complexity theory and the quantum marginal problem.

Facet-defining inequalities and geometric invariant theory play foundational roles: every moment polytope is cut out by finitely many facet inequalities that can be computed via support and orbit data, with error bounds reflecting the underlying representation parameters.

6. Structural Consequences and Future Directions

Moment polytopes encode, via convex geometry, the essence of representation-theoretic data, combinatorics, and the orbit structure of tensor varieties. Important consequences and avenues include:

• Grothendieck Semigroup Structure: The semigroup of spectral equivalence classes of representations under tensor product is isomorphic to the semigroup of Weyl-invariant convex integral polytopes under Minkowski addition; moment polytopes serve as complete invariants up to spectral equivalence (Kaveh et al., 2010).
• Intersection Theory: Kazarnovskii’s theorem provides a counting formula for solutions of systems defined by representation spaces, provably determined by mixed integrals of moment polytopes (Kaveh et al., 2010).
• Tensor Identifiability and Decomposition: Moment matrix extension algorithms transform tensor decomposition problems into linear algebraic equations when regularity is low, thereby relating the structure of the moment polytope to efficient computational procedures and identifiability thresholds (Shi et al., 27 Jun 2025).
• Explicit Examples and Non-freeness: The existence and explicit construction of non-free tensors demonstrates the limitations of Schmidt-type decompositions, with implications for quantum functionals and support functionals in asymptotic spectra (Berg et al., 28 Mar 2025).
• Real and Symplectic Geometry: Invariant-theoretic characterizations extend to real Hamiltonian manifolds via the identification of real moment polytopes as anti-invariant parts of Kirwan polytopes, parameterized by real Ressayre pairs and their facet equations (Paradan, 2020).

Future research directions include complete classifications in higher-dimensional tensor formats, precise relationships between degenerations/restrictions and polytope inclusions, and expanded algorithmic methods for symbolic and numerical computation in challenging cases. There is particular interest in leveraging algebraic geometry and combinatorial invariant theory to further clarify the operational landscape of moment polytopes of tensors, especially with respect to quantum entanglement, tensor network expressivity, and complexity-theoretic obstructions.