Strassen's Asymptotic Spectrum

• Strassen’s Asymptotic Spectrum is a duality-theoretic framework that characterizes asymptotic comparisons in preordered semirings using monotone homomorphisms.
• It establishes an asymptotic preorder via subexponential corrections and a duality theorem, equating spectral pointwise inequalities with resource convertibility.
• The framework underpins advances in tensor circuit complexity, graph invariants, and quantum functionals, offering practical insights into combinatorial optimization and resource theory.

Strassen’s asymptotic spectrum is a duality-theoretic and categorical framework for describing the asymptotic comparison of elements in preordered semirings, with deep applications to tensor analysis, graph invariants, algebraic complexity, and resource theories. It provides the universal domain for monotone semiring homomorphisms (spectral points), with the crucial property that asymptotic inequalities are equivalent to dual order in the spectrum. The program encompasses and unifies the analysis of computational resources (such as tensor rank), combinatorial optimizations, and various functional-analytic and convex-geometric invariants.

1. Preordered Semirings, Growth Conditions, and the Asymptotic Spectrum

Let $S$ be a commutative unital semiring with neutral elements $0,1$, equipped with a preorder $\leq$ satisfying $0\leq 1$ and compatibility with addition and multiplication. A monotone semiring homomorphism $f:S\rightarrow \mathbb{R}_{+}$ is a map preserving operations and order. The set of all such nonzero homomorphisms, denoted $\Delta(S,\leq)$, is the foundation of the asymptotic spectrum.

Strassen’s theory requires additional structure on $S$:

• The canonical inclusion $\mathbb{N}\hookrightarrow S$ is an order-embedding.
• There exists a “power-universal” $u\in S$, with $u\geq 1$ such that for every $x\neq 0$, some $k\in\mathbb{N}$ satisfies $x u^k\geq 1$ and $u^k x\geq 1$ ($S$ is then of polynomial growth).

These ensure that large tensor powers of elements can be compared up to subexponential correction, a necessary feature for operational resource theories and complexity models (Vrana, 2020, Fritz, 2018).

2. Asymptotic Preorder and Spectrum: Duality Theorem

The asymptotic preorder $\succeq$ on $S$ is defined via the existence of subexponentially negligible “padding” on large tensor powers: $$x \succeq y \iff \exists\,k_n: \, \lim_{n\uparrow\infty} k_n/n=0, \; \forall n:\; u^{k_n} x^n \geq y^n$$ This preorder is compatible with semiring operations and is independent of the specific choice of power-universal $u$.

The asymptotic spectrum $\Delta(S,\leq)$ is equipped with the coarsest topology making $f\mapsto f(x)$ continuous for every $x$, so that it is Tychonoff; if $S$ is generated by some $u$, the spectrum is locally compact.

The duality theorem (Strassen’s original, and its generalizations (Vrana, 2020, Fritz, 2018)) states:

• $\Delta(S,\leq)$ is (locally) compact Hausdorff.
• For all $x,y\in S$:

$$x \succeq y \iff \forall f\in \Delta(S,\leq): f(x)\geq f(y)$$

This translates asymptotic “convertibility” into pointwise dominance over all monotone homomorphisms (“spectral points”).

For polynomial growth semirings lacking a single power-universal element, a generalized spectrum is still constructed using a finite set $M\subset S$ and appropriate boundedness conditions (Vrana, 2020).

3. Strassen’s Positivstellensatz, Catalytic Inequalities, and Further Generalizations

Strassen’s Positivstellensatz provides equivalent algebraic and spectral characterizations of the asymptotic preorder. The original (“Archimedean”) variant required boundedness, which is weakened in generalizations to only polynomial growth (Fritz, 2018).

Let $S$ be a Strassen-preordered semiring. For nonzero $x, y \in S$, the following are equivalent:

• $f(x) \geq f(y)$ for all $f$ in the asymptotic spectrum.
• For every $\varepsilon > 0$, there exist $k, n \in \mathbb{N}$, $k \leq n$ such that $2^k x^n \geq y^n$. A more general form replaces the power term with arbitrary polynomials and introduces “catalytic” variables and padding (Fritz, 2018).

Several further equivalent “relaxed” conditions are provided. These generalizations accommodate resource-theoretic settings (such as quantum information theory) and semirings of functions or graphs with subtler growth dynamics.

4. Tensor Asymptotic Spectrum and Universal Spectral Points

A central application is to the semiring of tensors, with addition as direct sum, multiplication as tensor product, and the preorder encoding restriction (via multilinear maps). The asymptotic spectrum $\operatorname{Spec}_{\mathrm{asym}}(\mathcal{T})$ consists of all universal monotone semiring homomorphisms (“spectral points”) for tensors (Christandl et al., 2020, Sakabe et al., 29 Jan 2026).

Spectral points $\phi$ satisfy:

• Monotonicity under restriction,
• Multiplicativity and additivity,
• Normalization: $\phi(\text{unit tensor})=n$.

For tensors $t,s$, the duality theorem gives: $$t^{\otimes n+o(n)} \geq s^{\otimes n} \quad \Leftrightarrow\quad \forall\,\phi:\, \phi(t)\geq \phi(s)$$ Key examples include the asymptotic rank and subrank of tensors, quantum functionals, and support functionals.

A profound recent result is the identification of Strassen’s support functionals (defined combinatorially via support polytopes) with quantum functionals (defined via entropy optimization on entanglement polytopes), both providing universal spectral points (Sakabe et al., 29 Jan 2026).

5. Minimax Duality, Weighted Slice Ranks, and Entanglement Functionals

The study of asymptotic tensor invariants has led to minimax dualities uniting slice rank, support functionals, and quantum functionals. Weighted slice rank $S_\xi(T)$ generalizes the ordinary and non-commutative slice ranks by introducing weights $\xi$; its asymptotic growth

$$G_\xi(T) = \limsup_{n\to\infty} [S_\xi(T^{\otimes n})]^{1/n}$$

is a monotone functional and fits into the asymptotic spectrum framework.

A fundamental correspondence relates these functionals to quantum entropy-based invariants $F_\theta(T)$: $$G_\xi(T) = \min_{\theta \in \Theta(\xi)} F_\theta(T)^{1/\langle \theta, \xi\rangle}, \qquad F_\theta(T) = \max_{\xi \in \Xi} G_\xi(T)^{\langle \theta,\xi\rangle}$$ This minimax duality is established via convex-analytic techniques, with implications for rank-type characterizations of complexity-theoretic quantities and for the extension of quantum invariants to arbitrary fields (Christandl et al., 2020, Sakabe et al., 29 Jan 2026).

Recent work has confirmed that, for all probability vectors $\theta$, Strassen’s support functional $\zeta^\theta(T)$ equals the quantum functional $F_\theta(T)$, using minimax duality over polytopes and Fenchel-type duality on Hadamard manifolds (Sakabe et al., 29 Jan 2026).

6. Applications: Circuit Complexity, Graph Invariants, and Algorithmic Barriers

Strassen's asymptotic spectrum provides powerful obstructions and dual characterizations for several computational problems:

• Matrix/Tensor Circuit Complexity: For matrices $M$, the asymptotic spectrum equates the optimal depth-2 linear circuit size $\sigma(M)$ with the supremum over “$\alpha$-volume” spectral points, parameterized by $\alpha\in[0,1]$ (Alman et al., 17 Sep 2025). The best possible exponent for disjointness matrices is $\sigma(R) < 2^{1.249424}$, providing improved explicit circuit constructions.
• Algorithmic Applications: Optimal circuits for Kronecker powers of $M$ enable fast algorithms for the Orthogonal Vectors problem, as the circuit size and maximum degree control the running time bounds.
• Graph Theory: The spectrum captures quantities like the Shannon capacity of graphs, with spectral points including Lovász’s theta number and the fractional chromatic number. The duality equates asymptotic comparisons to pointwise spectral inequalities (Fritz, 2018).

This framework has also been extended to function semirings, polynomial semirings, and resource-monotone invariants in quantum Shannon theory.

7. Further Directions and Open Problems

Key open problems include:

• Determining under what conditions catalytic variables or “padding” can be eliminated in the generalized Positivstellensatz (Fritz, 2018).
• Generalizing the support and quantum functionals to completely algebraic or purely functional-analytic domains.
• Exploring local compactness or Choquet simplex structure of the spectrum in broad classes.
• Applying minimax and convex-geometric dualities to new parameters in resource theory and tensor complexity, specifically in contexts where moment and support polytopes correspond.

The ongoing investigation of the asymptotic spectrum connects functional analysis, tensor algebra, combinatorics, and complexity theory, establishing a deep and unifying language for asymptotic comparison of mathematical resources (Vrana, 2020, Fritz, 2018, Christandl et al., 2020, Sakabe et al., 29 Jan 2026, Alman et al., 17 Sep 2025).