Entanglement Polytopes in Quantum Systems

• Entanglement polytopes are convex geometric objects that represent all possible local eigenvalue spectra obtainable from SLOCC transformations of a pure multipartite quantum state.
• They can be computed using covariant-based methods and moment map techniques, providing practical algorithms for entanglement detection in small to moderate quantum systems.
• Their structure reveals the hierarchy of multipartite entanglement classes and aids in solving the quantum marginal problem and understanding tensor complexity.

Entanglement polytopes are convex polytopes arising from the set of possible single-particle reduced-state spectra attainable within the closure of a stochastic local operations and classical communication (SLOCC) orbit of a pure multipartite quantum state. They provide a finite, coarse-grained classification of multipartite entanglement based solely on local eigenvalue data, furnishing a bridge between geometric invariant theory, symplectic geometry, and practical entanglement detection.

1. Mathematical Definition and Theoretical Framework

Let $\mathcal{H} = \bigotimes_{i=1}^N \mathbb{C}^{d_i}$ be the Hilbert space of an $N$-partite quantum system. Given a pure state $|\psi\rangle \in \mathcal{H}$, denote by $$\rho^{(i)} = \operatorname{Tr}_{j \neq i} |\psi\rangle\langle\psi|$$ the one-body reduced density matrix for subsystem $i$. The ordered eigenvalues $$\lambda^{(i)} = (\lambda^{(i)}_1 \geq \cdots \geq \lambda^{(i)}_{d_i})$$ form the local spectra.

Entanglement polytopes arise as the set of all local spectra that can be obtained from the closure $\overline{G \cdot [\psi]}$ of a SLOCC orbit $G \cdot [\psi]$ in projective space $\mathbb{P}(\mathcal{H})$, where $$G = \operatorname{SL}(d_1,\mathbb{C}) \times \cdots \times \operatorname{SL}(d_N,\mathbb{C})$$. Explicitly,

$$\Delta_{[\psi]} := \left\{ (\lambda^{(1)},\dots, \lambda^{(N)}) : \exists \phi \in \overline{G \cdot [\psi]},~ \operatorname{Spec}(\rho^{(i)}_\phi) = \lambda^{(i)} \right\}.$$

By the convexity theorems of Kirwan, Brion, and Guillemin–Sternberg, $\Delta_{[\psi]}$ is a compact, rational convex polytope in $\prod_i \Delta_{d_i}$, where $\Delta_{d_i}$ is the probability simplex of spectra for system $i$ (Walter et al., 2012, Berg et al., 9 Oct 2025, Hirai, 15 Nov 2025, Bürgisser et al., 2018, Wernli, 2018). Invariant-theoretically, the vertices correspond to normalized highest weights of nonvanishing covariants on $[\psi]$, realized as

$\Delta_{[\psi]} = \overline{\big\{ \tfrac{1}{n} (\mu^{(1)},\dots,\mu^{(N)}) : \text{there is a degree-%%%%15%%%% %%%%16%%%%-covariant of weight } (\mu^{(1)},\dots,\mu^{(N)}) \neq 0 \text{ on } [\psi] \big\}}.$

2. Relation to SLOCC Entanglement and Moment Maps

The SLOCC group $G$ partitions pure states into orbits corresponding to different entanglement types. The inclusion hierarchy of orbit closures induces an inclusion of polytopes: if $$\overline{G \cdot [\psi]} \subset \overline{G \cdot [\phi]}$$ then $\Delta_{[\psi]} \subset \Delta_{[\phi]}$. The structure of entanglement polytopes thus reflects the hierarchy of multipartite entanglement (Walter et al., 2012, Wernli, 2018).

The entanglement polytope can be equivalently realized as the image, under the moment map $\mu$, of the projective $G$-orbit of $[\psi]$, intersected with the positive Weyl chamber. For $k$-partite tensor spaces, this reads

$$\Delta(T) = \mu( \overline{G \cdot [T]} ) \cap \mathcal{W}^+,$$

where $\mathcal{W}^+$ denotes the set of ordered spectra (Berg et al., 9 Oct 2025, Hirai, 15 Nov 2025, Bürgisser et al., 2018).

3. Algorithmic Computation and Scaling Methods

Two principal algorithmic paradigms exist for computing entanglement polytopes:

• Covariant-based methods: Generate all relevant $G$-covariants, retain those nonvanishing on a representative $|\psi\rangle$, and form the convex hull of the corresponding normalized weights (Walter et al., 2012, Wernli, 2018). This approach is computationally tractable only for modest system sizes due to the rapid growth of the covariant algebra.
• Moment map and support techniques: Exploit Franz's convex-geometric description, in which the polytope is the intersection of convex hulls of the weights obtained under lower-triangular Borel group actions on the tensor, and the Weyl chamber (Berg et al., 9 Oct 2025). This formulation underpins a symbolic Gröbner-basis-based method as well as probabilistic and heuristic membership testing via tensor scaling (Bürgisser et al., 2018). In practice, the tensor-scaling algorithm enables polynomial-time weak membership oracles for entanglement polytopes, simultaneously certifying non-membership and approximate realization of target spectra.

A summary of computational features:

Method	System Size Feasibility	Key Resources
Covariant (GIT)	3–4 qubits/low-dim	Invariant-theory
Franz/Support Tests	up to $4 \times 4 \times 4$	Gröbner bases, scaling
Tensor Scaling (PTIME)	Arbitrary (membership)	Cholesky + marginals

4. Examples and Hierarchy in Low Dimensions

In three qubits ($d_1 = d_2 = d_3 = 2$), the entanglement polytopes are explicitly classified (Walter et al., 2012, Zhao et al., 2015, Berg et al., 9 Oct 2025):

• Separable class: single point $(1,1,1)$.
• Biseparable classes: line segments connecting $(1,1,1)$ to $(1/2,1/2,1)$, etc.
• W class: the set $$\{\lambda \in [1/2,1]^3 : \lambda_1 + \lambda_2 + \lambda_3 \geq 2\}$$.
• GHZ class: the full "possible spectra" polytope, determined by the qubit marginal constraints.

For $N=3$, the W-polytope sits strictly inside the GHZ-polytope, and Haar-random pure states have local spectra within the W-polytope with probability $\approx 94\%$ (Zhao et al., 2015). For higher $N$, the number of distinct polytopes remains finite, but their pairwise overlaps increase dramatically, degrading the resolving power of local spectra alone (Maciążek et al., 2017).

5. Experimental Detection and Overlap Issues

Due to overlaps among polytopes, local spectra alone typically provide only limited information about the entanglement class, especially for $N \geq 4$ (Zhao et al., 2015, Maciążek et al., 2017). For example, in generic four-qubit pure states, nearly all Haar-random states share local spectra within the largest polytope, rendering single-shot spectral witness ineffective.

To mitigate these degeneracies, local filter operations—nonunitary local transformations such as projective measurements or amplitude damping—can be used to "move" the local-spectra vector within the polytope. By monitoring how the filtered spectra traverse the inequalities defining different entanglement polytopes, one can unambiguously distinguish between otherwise degenerate entanglement classes, as experimentally demonstrated for four-qubit photonic systems (Zhao et al., 2015). This approach requires only local operations and single-qubit tomography, avoiding the exponential overhead of full state tomography.

6. Asymptotic Properties for Large Systems

For large $N$, most full-dimensional entanglement polytopes accumulate near the maximally mixed point in the spectra hypercube, with distances scaling as $1/(2\sqrt{N})$ (Maciążek et al., 2017). The volume of their mutual overlap can become extensive, rendering entanglement polytopes increasingly ineffective as global entanglement witnesses based on local eigenvalue data. Moreover, even after optimal SLOCC distillation (gradient flow to the maximally mixed local states), the limiting spectra converge to this congested region, so class separation cannot be improved by distillation protocols. Noise robust witnessing requires purity $1 - O(1/N)$, often beyond current experimental capabilities.

7. Applications and Computational Advances

Entanglement polytopes underpin the solution to the one-body quantum marginal problem, specifying which tuples of local spectra can occur for pure states given local dimensions (Bürgisser et al., 2018, Berg et al., 9 Oct 2025). They also play a central role in algebraic complexity theory (e.g., Kronecker polytopes, subrank bounds for matrix multiplication) and quantum resource theory, providing geometric invariants for tensors and criteria for LOCC/SLOCC reachability.

Recent algorithmic progress, including representation-theoretic scaling algorithms and convex optimization over polytopes via generalized gradient flows on Hadamard manifolds, has enabled weak membership oracle construction and numerical computation of high-dimensional entanglement polytopes up to the $4 \times 4 \times 4$ tensor case (Berg et al., 9 Oct 2025, Hirai, 15 Nov 2025). Explicit lists of facet inequalities for $3 \times 3 \times 3$ and $4 \times 4 \times 4$ polytopes have been compiled. The structure of entanglement polytopes provides insight into tensor non-freeness, marginal constraints, and scaling algorithm runtime constants.

Ongoing challenges concern optimal filter strategies for experimental detection, noise robustness, extension to mixed states, higher-dimensional subsystems, and the full hierarchy of polytopes in the large-system limit.

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References:

• (Walter et al., 2012) Entanglement Polytopes: Multiparticle Entanglement from Single-Particle Information
• (Zhao et al., 2015) Experimental Detection of Entanglement Polytopes via Local Filters
• (Maciążek et al., 2017) Asymptotic properties of entanglement polytopes for large number of qubits
• (Bürgisser et al., 2018) Efficient algorithms for tensor scaling, quantum marginals and moment polytopes
• (Wernli, 2018) Computing Entanglement Polytopes
• (Berg et al., 9 Oct 2025) Computing moment polytopes – with a focus on tensors, entanglement and matrix multiplication
• (Hirai, 15 Nov 2025) Generalized gradient flows in Hadamard manifolds and convex optimization on entanglement polytopes