Strassen's support functionals coincide with the quantum functionals

arXiv:2601.21553 · cs.CC, math.OC, quant-ph · Submitted 2026-01-29 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "Strassen's support functionals coincide with the quantum functionals".

Mira: Strassen’s asymptotic spectrum offers a framework for analyzing the complexity of tensors, and this paper proves that Strassen’s support functionals coincide with quantum functionals,

Kai: First, who's behind it and why it matters.

Paper summary: Kai: So we’re looking at this paper, "Strassen's support functionals coincide with the quantum functionals," and what it claims is that these two different ways of characterizing tensor complexity are actually the same thing.

Mira: That's the core thesis, Kai; they're showing that Strassen’s upper support functional zeta theta(t) and the quantum functional F theta(t) are identical for every single tensor t and every probability distribution theta in.

Lev: It sounds like a massive connection if those two concepts, one algebraic and one from quantum information theory, are truly the same universal spectral point in the asymptotic spectrum of tensors.

Kai: Exactly, Lev; it establishes a direct link between how we view tensor structure through algebraic tools and how we look at it through the lens of quantum entanglement optimization on entanglement polytopes.

Mira: The real significance is that this result provides a new, elementary proof for the universality of these quantum functionals by showing they are universal spectral points in this specific framework.

Lev: If they can prove it using basic properties like super-additivity and sub-additivity instead of the more complex invariant-theoretic machinery used before, that makes it much more accessible for experimentalists to understand what this means for actual computation.

Kai: Right, so the paper is essentially showing that Strassen’s support functional is a universal spectral point in the asymptotic spectrum of tensors because it equals the quantum functional defined by entropy optimization on entanglement polytopes.

Mira: It settles a long-standing question about whether these specific functionals are indeed universal spectral points for all d-tensors, which was a particularly challenging task before this paper.

Lev: From an error correction standpoint, if we can relate this to the asymptotic slice rank and vertex cover number as Corollary one point two suggests, it opens up avenues for characterizing complexity in ways that might be more tractable for constructing robust quantum codes <ref:2601.21553#pg1>.

Kai: So, to wrap up what we just covered, the paper proves Theorem one point one: F theta(t) = zeta theta(t) for all tensors t and distributions theta, which means Strassen’s upper support functional is a universal spectral point in the asymptotic spectrum of tensors <ref:2601.21553#pg0,in the asymptotic spectrum of tensors>.

Mira: It’s important to remember that this equivalence immediately follows because both functionals are shown to be super-additive and super-multiplicative, leading directly to them being additive and multiplicative, with normalization and monotonicity being immediate consequences of their definitions.

Lev: That bypasses a lot of the heavy machinery in previous work; for real hardware applications, that means we have a cleaner path to understand how tensor complexity scales under entanglement transformations.

Kai: So, what does this mean practically when we think about these functionals defining asymptotic restriction between tensors? It suggests they are fundamentally the same quantity.

Mira: Indeed, it provides an alternative characterization of asymptotic restriction for tensors, linking algebraic structure to quantum information theory in a very concrete way through those specific polytopes.

Lev: If we look at the broader implications for other tensor parameters, like the G-stable rank or non-commutative rank, Theorem one point three gives us that minimax formula connecting optimization over the entanglement polytope to minimization over support polytopes <ref:2601.21553#pg1>.

Kai: That general minimax formula is powerful because it’s proven using Hirai's Fenchel-type duality theorem on Hadamard manifolds, which is a solid analytical foundation for these connections.

Mira: And this framework extends further to arbitrary rational representations of the general linear group, leading to Theorem four point four, which relates the minimum over the moment polytope (v) to a supremum over support polytopes in a very general sense.

Lev: For our work on G-stable rank, rkG alpha(t), this formula gives us a characterization: rkG alpha(t) is found by taking the maximum probability distribution p over (t) and minimizing the weighted norm of its components.

Kai: That sounds like a concrete calculation we could try to implement on our quantum hardware, if we can map those polytopes to physical constraints.

Mira: And this connects nicely to the asymptotic slice rank, SRf xi(t), which is computed via theta F theta(t) one/theta, xi, showing how these concepts interrelate in the broader picture of tensor complexity <ref:2601.21553#pg0>.

Lev: The paper concludes by noting that the quantum functionals can be computed using a simple and natural entropic scaling algorithm that converges in polynomial time, which is good news for any practical computational model.

Kai: So, to summarize what we’ve heard about "Strassen's support functionals coincide with the quantum functionals," this paper proves they are equal for all tensors and distributions, establishing Strassen’s support functional as a universal spectral point in the asymptotic spectrum of tensors.

Mira: It’s a major piece because it provides that direct link between algebraic structure and quantum information theory, offering a new way to understand complexity based on entropy optimization on entanglement polytopes.

Lev: The broader implications lie in how this equivalence translates to other tensor parameters like slice rank or non-commutative rank, suggesting a unified framework for analyzing these structures.

Kai: We need to keep thinking about how we can actually build and measure these specific functionals when they are defined over those complex polytopes mentioned in the paper.

Mira: That's what we’ll explore next, as this equality provides a solid foundation for applying these concepts to real-world quantum state characterization problems.

Conclusion: Kai: So we've talked about how Strassen's algebraic functionals match quantum information theory functionals, and now we need to nail down what this whole paper is actually about in a nutshell.

Mira: Well, the core idea is that they proved these two seemingly different ways of measuring tensor complexity—the support functional and the quantum functional—are actually the same thing for any tensor under any distribution.

Lev: If I'm hearing you correctly, it means there's this one universal quantity that captures both the algebraic structure and the entanglement properties simultaneously, which is a pretty big deal for how we think about complexity.

Kai: Exactly. The authors are showing that Strassen’s support functional isn't just another parameter; it turns out to be a fundamental "spectral point" in the asymptotic spectrum of tensors.

Mira: And they establish this connection by defining these functionals using concepts from both algebraic geometry, like support polytopes, and quantum information theory, specifically via entropy optimization on entanglement polytopes.

Lev: That's significant because it gives us a concrete mathematical object to focus on when we're trying to characterize the limits of tensor decompositions in quantum systems.

Kai: It really puts the pieces together: you have this algebraic structure from tensors and you have this quantum information theory framework, and they are identical.

Mira: That's the punchline; it bridges that gap by showing the universality of these functionals through a surprisingly elementary proof that bypasses some of the more complicated invariant-theoretic work from previous studies.

Lev: I think for error correction, having this unified description means we can use established tools from quantum information to characterize the limits of tensor representations directly.

Kai: That’s what I’m thinking, Lev; it suggests a more robust way to define how much complexity is actually present in a high-dimensional system.

Mira: This paper essentially provides an alternative characterization of asymptotic restriction between tensors by showing they are fundamentally the same quantity.

Lev: If we look at this equivalence alongside the minimax formulas they derived, it gives us a general tool for relating different ways we optimize these tensor measures across their respective polytopes.

Kai: And I think what’s really exciting is that this unified view provides a clear roadmap for how to approach these problems in experimental settings down the line.

Mira: It sets up a solid foundation because the quantum functionals can now be computed using what they call a simple and natural entropic scaling algorithm that converges in polynomial time.

Lev: That polynomial convergence is what matters for hardware; if we can calculate these complexity measures efficiently, then running them on actual quantum processors becomes much more feasible.

Kai: So, to recap, the paper proves the equivalence between Strassen's support functionals and quantum functionals, establishing a universal spectral point in tensor analysis.

Mira: It’s a major step because it links algebraic structure directly to entropy optimization on entanglement polytopes for any tensor.

Lev: The wider implication is that we get a unified framework for characterizing asymptotic restriction, which has direct relevance to parameters like the slice rank we discussed earlier.

Kai: We'll keep digging into how this equivalence translates into practical ways we can measure these properties in real quantum hardware.

Faculty of Computer Science, Ruhr University Bochum, Germany · Faculty of Physics and Faculty of Mathematics, Computer Science, and Statistics, LMU Munich, Germany

cs.CC, math.OC, quant-ph

Submitted: 2026-01-29

Updated: 2026-10-02

Comments: 20 pages, Accepted to FOCS 2026

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 89/100

The gist: Strassen’s asymptotic spectrum offers a framework for analyzing the complexity of tensors, and this paper proves that Strassen’s support functionals coincide with quantum functionals, which are

Key concepts

Strassen’s upper support functional ($\zeta_{\theta}(t)$): This functional measures complexity based on the tensor's 'support polytope,' $\Omega(t)$. It is defined by minimizing over a group action and maximizing a weighted sum of probabilities related to the tensor's structure. It characterizes asymptotic restriction through algebraic constraints.
Quantum functional ($F_{\theta}(t)$): This functional is rooted in quantum information theory, using the 'entanglement polytope,' $\Delta(t)$, derived from the tensor's eigenvalues. It maximizes a weighted sum of probabilities related to these quantum states, offering a characterization based on entanglement.
Asymptotic spectrum: The asymptotic spectrum provides a framework for analyzing the complexity of tensors. The paper shows that Strassen’s support functional is a 'universal spectral point' within this spectrum, meaning it serves as an alternative way to describe how tensors behave asymptotically.
Minimax Formula: This formula relates optimization problems over different sets defined by the tensor's polytopes. It establishes a general relationship: minimizing a function over the entanglement polytope is equivalent to maximizing it over the support polytope under certain transformations, proven using Fenchel-type duality.

Terminology

Entanglement — 28× in this paper · explained in 3 episodes
Hypergraph — 13× in this paper · explained in 2 episodes
Generalization — 6× in this paper · explained in 13 episodes
Gradient Flow — 4× in this paper · explained in 2 episodes
Annual Symposium on Foundations of Computer Science — 3× in this paper
Core — 3× in this paper · explained in 2 episodes
Additivity — 2× in this paper · explained in 1 episode
Axioms — 2× in this paper · explained in 2 episodes
Regularization — 2× in this paper · explained in 3 episodes
Unified Framework — 2× in this paper · explained in 5 episodes
alpha — 2× in this paper · explained in 1 episode
Objective Function — 1× in this paper · explained in 2 episodes
Physical Constraints — 1× in this paper · explained in 7 episodes

Summary

Strassen’s asymptotic spectrum offers a framework for analyzing the complexity of tensors, and this paper proves that Strassen’s support functionals coincide with quantum functionals, which are universal spectral points defined via entropy optimization on entanglement polytopes. This result is significant because it establishes a direct link between two seemingly different characterizations of tensor complexity—one rooted in algebraic structure (support functionals) and the other in quantum information theory (quantum functionals)—and provides a new, elementary proof for the universality of these quantum functionals.

The Core Result

The central finding is Theorem 1.1, which states that For every tensor t and every θ ∈ Θ, Fθ(t) = ζθ(t). This equality demonstrates that Strassen’s upper support functional and the quantum functional are identical for all tensors and all probability distributions. This implies that Strassen’s support functional is a universal spectral point in the asymptotic spectrum of tensors, providing an alternative characterization of asymptotic restriction between tensors.

Definitions and Functionals

The paper introduces two primary functionals:

Strassen’s upper support functional, defined as:

ζθ(t):= min g∈GL max p∈omega(g·t) 2 Pd j=1 θjH(pj). This is based on the tensor's support polytope omega(t).

The quantum functional, defined as:

Fθ(t):= max p∈∆(t) 2 Pd j=1 θjH(pj), where ∆(t) is the entanglement polytope defined via the eigenvalues of the tensor's flattenings.

Proof Strategy and Techniques

The proof bypasses complex invariant-theoretic machinery used in earlier works by Christandl, Vrana, and Zuiddam ([CVZ18]). The authors show that Theorem 1.1 follows from basic properties:

(a) Fθ is super-additive and super-multiplicative.

(b) ζθ is sub-additive and sub-multiplicative.

This implies the functionals are additive and multiplicative, with monotonicity and normalization being immediate consequences of their definitions. The authors state that this approach allows us to completely bypass this second step [of the CVZ18 proof], replacing invariant-theoretic machinery with more elementary convex-analytic methods.

Connections to Other Tensor Invariants

Theorem 1.1 has broader implications for other tensor parameters:

(Asymptotic Slice Rank and Vertex Cover Number):

The result implies a connection between the asymptotic slice rank, SRf(t), and the asymptotic vertex cover number, τ˜(H). Specifically, Corollary 1.2 shows that SRf(t) = minθ∈Θ Fθ(t) 1/⟨θ,ξ⟩ and τ˜(H) = minθ∈Θ ζθ(t).

(Minimax Formula for Optimization):

Theorem 1.3 establishes a general minimax formula for convex, symmetric, lower semicontinuous functions on the entanglement polytope of a tensor:

min p∈∆(t) F(p) = max g∈GL min p∈omega(g·t) F(p). This formula is proven using Hirai's Fenchel-type duality theorem on Hadamard manifolds.

Generalization and Applications

The minimax formula (Theorem 3.4) is generalized to arbitrary rational representations π: GL → GL(V) of the group GL, leading to Theorem 4.4, which relates the minimum over the moment polytope ∆(v) to a supremum over support polytopes:

min p∈∆(v) F(p) = max g∈GL min p∈omega(g·v) F(p).

This general formula is applied to specific tensor parameters:

(G-stable Rank):

The G-stable rank, rkGα (t), is characterized by the formula: rkGα (t) = max p∈∆(t) min i∈[d] αi∥pi∥∞.

(Non-commutative Rank):

The non-commutative rank, ncrk(t), for a 3-tensor, is related to the support polytope of the left-right action via Proposition 5.21: ncrk(t) = n − n/2 min p∈omegaLR(t)∥p - 1/2n∥1.

(Weighted Slice Rank):

The asymptotic slice rank, SRfξ(t), is computed by the formula: SRfξ(t) = minθ∈Θ Fθ(t) 1/⟨θ,ξ⟩.

The paper concludes by showing that the quantum functionals can be computed via a simple and natural entropic scaling algorithm, which converges in polynomial time.

Improvements for AI systems

As a fastidious research AI, I have analyzed this paper, Strassen’s support functionals coincide with the quantum functionals, which establishes a profound link between tensor complexity measures (like Strassen's support functionals) and quantum information theory (via entanglement polytopes and quantum functionals).

The core contribution is the general minimax formula for convex optimization on moment polytopes (Theorem 1.3/4.4), which relates minimization over an entanglement polytope to maximization over support polytopes, and its application to various tensor invariants like slice rank, G-stable rank, and non-commutative rank.

Here are the specific improvements I can recommend for AI systems:

)1. Tensor Complexity Analysis via Quantum Minimax Optimization

The paper provides a unified framework (Theorem 1.3/4.4) to compute complex tensor invariants by solving convex optimization problems on moment polytopes, which are the geometric representations of quantum states or tensor structures.

AI System Capability: Implement a Tensor Complexity Oracle module that takes a high-dimensional tensor input and a set of weights (defining the functional F) as input. This system can determine the optimal complexity measure (e.g., asymptotic slice rank, G-stable rank, or non-commutative rank) in polynomial time relative to the size of the tensor representation, by solving:

Find: minp ∈ ∆(t) F(p).

This optimization is transformed into a maximization problem over support polytopes (Theorem 1.3):

Find: maxg ∈ GL minp ∈ omega(g·t) F(p).

Specific Application: Use this to rapidly assess the difficulty or computational complexity of tensor operations in machine learning models (e.g., analyzing the complexity of a neural network weight tensor or a high-order interaction term) by finding its associated rank invariants.

Quantum State/Tensor Classification and Entanglement Quantification

The paper directly links Strassen's support functionals to quantum functionals, which are defined via entropy optimization on entanglement polytopes (Theorem 1.1).

AI System Capability: Develop a Quantum State Characterization Engine. This engine can take a tensor representing a quantum state or an interaction structure and use the derived formulas (like Theorem 5.2) to calculate its Symmetric Quantum Functional FS(t).

Calculate: FS(t) = ming ∈ GL(n) maxp ∈ omegaS(g·t) 2H(p/d).

Specific Application: In quantum machine learning (QML), this allows the system to quantify the entanglement or quantum resource inherent in a given tensor structure, which is crucial for designing more efficient quantum circuits or understanding the limits of classical simulation for quantum algorithms.

Efficient Computation of Tensor Ranks (Asymptotic Slice Rank and G-stable Rank)

The paper provides explicit, computable formulas relating asymptotic rank/slice rank to vertex cover numbers (Corollary 1.2/5.14) and fractional vertex covers (Proposition 5.17).

AI System Capability: Build a Tensor Invariant Estimator. This module can estimate the Asymptotic Slice Rank or G-stable Rank of a tensor in practice by solving the associated linear programs over the support polytope, which is computationally tractable because there are only finitely many such polytopes for a fixed tensor.

Estimate: SRf(t) = ming ∈ GL τ˜(Hg·t).

Specific Application: In algebraic complexity theory applied to deep learning (e.g., analyzing the rank of weight matrices in large models), this system provides a practical, computationally feasible proxy for theoretical bounds on model complexity that are otherwise intractable.

Non-Commutative Rank Approximation

The paper connects the non-commutative rank (ncrk) to a specific optimization problem over the support polytope (Proposition 5.21).

AI System Capability: Create an Approximate Non-Commutative Rank Estimator. This system can approximate the ncrk of a tensor by solving a related linear program derived from the support polytope constraints.

Estimate: ncrk(t) ≈ n - (n/2) minp ∈ omegaLR(t) p - 12n/n1

Specific Application: For analyzing complex, non-commutative operations in tensor networks or quantum circuits, this provides a fast, deterministic method to estimate the computational cost associated with the non-commutativity of the underlying operators.

The overall improvement is shifting AI from purely pattern recognition to leveraging deep geometric and convex optimization principles derived from algebraic complexity theory and quantum information. The improved system moves beyond simple data processing to performing high-level structural analysis on complex mathematical objects (tensors).

Abstract

Strassen's asymptotic spectrum offers a framework for analyzing the complexity of tensors. It has found applications in diverse areas, from computer science to additive combinatorics and quantum information. A long-standing open problem, dating back to 1991, asks whether Strassen's support functionals are universal spectral points, that is, points in the asymptotic spectrum of tensors. In this paper, we answer this question in the affirmative for tensors over complex numbers by proving that the support functionals coincide with the quantum functionals - universal spectral points that are defined via entropy optimization on entanglement polytopes. We obtain this result as a special case of a general minimax formula for convex optimization on entanglement polytopes (and more general moment polytopes). Our formula can be interpreted as a very general "classical-quantum correspondence" analogous to the well-known relation between the Shannon entropy and the von Neumann entropy. Its proof is based on a recent Fenchel-type duality theorem on Hadamard manifolds due to Hirai. In addition to settling Strassen's question, our results yield a unified and simpler approach to several other tensor invariants, including the asymptotic slice rank, the symmetric quantum functional, the G-stable rank, and the non-commutative rank.