Higher Semi-Additivity: Quantum & Tensor Perspectives

• Higher semi-additivity is a framework that generalizes classical additivity by allowing resource values to scale via a function of select lower-copy measures rather than linearly.
• It utilizes companion matrix recurrences to derive scaling exponents for quantifiers in quantum resource theories and tensor border rank, revealing subadditive and super-additive regimes.
• This concept has practical implications in optimizing tensor decompositions and computational complexity, impacting areas such as matrix multiplication and quantum state resource analysis.

Higher semi-additivity, also termed augmented additivity or $q$-additivity, generalizes the classical concept of additivity in the quantitative theory of resources, with substantial ramifications in quantum information theory, tensor analysis, and the study of computational complexity. In contrast to strict additivity, where a quantifier’s value scales linearly or additively with respect to independent objects (such as copies of a quantum state or direct sums of tensors), higher semi-additivity allows for a broader class of linear and sublinear scaling relations. This concept finds precise formalization both in quantum resource theories—through the behavior of entanglement, coherence, or analogous monotones—and in the algebraic study of tensors, where the border rank embodies nontrivial departures from naive additive behavior for higher-order objects.

1. Formal Definition and Conceptual Framework

Let $\varrho\in\mathcal{B}(\mathcal{H})$ be a quantum state and ${\cal E}(\varrho^{\otimes N})$ a resource quantifier (entanglement, coherence, etc.). Traditional additivity prescribes $${\cal E}(\varrho^{\otimes N}) = N\,{\cal E}(\varrho)$$. Higher semi-additivity generalizes this by postulating the existence of integers $1 \leq i_1 < i_2 < \cdots < i_q < N$ such that ${\cal E}(\varrho^{\otimes N})$ is determined entirely by the tuple $${\vec{e}} = ({\cal E}(\varrho^{\otimes i_1}),\dots,{\cal E}(\varrho^{\otimes i_q}))$$ via a function $E^{(N)}$: ${\cal E}(\varrho^{\otimes N}) = E^{(N)}(\vec{e}).$ If $E^{(N)}$ is a linear map, there exists a weight vector ${\vec{N}}$ such that

$${\cal E}(\varrho^{\otimes N}) = {\vec{N}}\cdot{\vec{e}}.$$

The minimal $q$ for which this holds is termed the $q$-scalability of the quantifier, and the property is $q$-additivity or augmented additivity. The domain of $N$ is typically restricted to powers of a fixed base $a$: $N \in \mathds{P}_a = \{a^n: n\in\mathbb{N}\}$, but empirical evidence shows broader applicability (Melo et al., 2022).

2. Border Rank and Semi-Additivity in Tensors

The classical paradigm of additivity is epitomized by the matrix rank under direct sum, where $R(T \oplus S) = R(T) + R(S)$ always holds. However, in the context of higher-order tensors ($k \geq 3$), both the tensor rank $R(T)$ and border rank $\underline{R}(T)$ can fail to be additive, exhibiting strict subadditivity or semi-additivity: $$\underline{R}(T \oplus S) < \underline{R}(T) + \underline{R}(S).$$ Schönhage provided explicit $k=3$ counterexamples in 1981. For $k \geq 4$, recent constructions establish that such failures are not incidental but structural; for any $k \geq 4$ there exist pairs $T_1,T_2$ such that

$$\underline{R}(T_1 \oplus T_2) \leq \underline{R}(T_1) + \underline{R}(T_2) - \delta$$

for some $\delta > 0$ (Christandl et al., 2020). These strict inequalities, termed “semi-additivity” in the border rank context, have concrete computational implications, particularly concerning the complexity of matrix multiplication and the exponents of more general graph tensors.

3. Structural and Algorithmic Construction

The analytic backbone of higher semi-additivity in resource theory is the derivation and solution of linear recurrence relations based on the scaling of small-copy data. For a $q$-additive quantifier, the weights ${\vec{N}}$ are computed recursively via a $q \times q$ companion matrix $\mathcal{Q}$. If the recurrence is

$$\eta^{i_j}(N) = \sum_{\ell=1}^q \eta^{i_j}(i_\ell K)\;\eta^{i_\ell}(N/K)$$

for $N = K a^n$, diagonalization of $\mathcal{Q}$ determines the scaling exponents $\nu_k = \log_{a} \lambda_k$, where $\lambda_k$ are the eigenvalues of $\mathcal{Q}$. The general solution reads

$$\eta^{i_j}(a^n) = \sum_{k=1}^q C^{i_j}_k (a^n)^{\nu_k},$$

with constants fixed by boundary conditions. For $q=1$, ordinary additivity is recovered; for $q \geq 2$, the explicit formulas interpolate between sub-additive, super-additive, and weakly additive regimes, depending on the real parts of $\nu_k$ (Melo et al., 2022).

In tensor theory, strict subadditivity of border rank is demonstrated through explicit constructions leveraging the flattening characterization of $\underline{R}$, polynomial families of rank-one tensors in an auxiliary parameter $\epsilon$, and limit arguments that eliminate intermediate terms, isolating only the extremal (constant and highest-degree) contributions. For example, for an order-4 spider and matrix pair,

$$\underline{R}(T_1) = (n_1 + 1)(n_2 + 1)(n_3 + 1),\quad \underline{R}(T_2) = N,$$

but

$$\underline{R}(T_1 \oplus T_2) = (n_1 + 1)(n_2 + 1)(n_3 + 1) + 1,$$

with $\delta = N-1$ (Christandl et al., 2020).

4. Interpolation Between Additive, Semi-Additive, and Super-Additive Behavior

An essential feature of higher semi-additivity is its capacity to interpolate among various scaling regimes. For $q=2$, consider the recurrence associated with a $2\times2$ companion matrix with eigenvalues $\lambda_{1,2}$. The scaling of ${\cal E}(\varrho^{\otimes N})$ is determined by the relative values of $\nu_1, \nu_2$:

• Both $\nu_1, \nu_2 < 1$: strict sub-additivity (${\cal E}(\varrho^{\otimes N})/N \to 0$),
• Either $\nu_k > 1$: super-additive divergence,
• One $\nu_k = 1$, others $<1$: weak additivity (finite slope at infinity).

This framework naturally accommodates resource monotones that are not strictly additive but obey higher-order scaling laws, providing a systematic approach to quantify and utilize these departures (Melo et al., 2022).

5. Applications and Explicit Examples

The $q$-additive formalism is applicable to diverse domains:

• Quantum Resource Theory: Empirical accuracy is demonstrated for one-shot PPT-distillable entanglement of isotropic states, leveraging a three-term scaling law with exponents corresponding to $N$, $\sqrt{N}$, and a constant. For isotropic states in dimension $d \otimes d$, the predicted scaling matches numerical linear-programming results within $10\%$ for moderate $N$ (Melo et al., 2022).
• Tensor Complexity: In higher-order tensor constructions (spider plus edge or GHZ structures), explicit calculations provide the border-rank deficit $\delta$ compared to naive additivity. For instance, in a $4 \times 4 \times 4$ spider, $\delta=3$; for an order-4 multi-leg construction with $n=10$, $\delta \geq 699$ out of a trivial bound of $2000$ (Christandl et al., 2020).
• Asymptotic Rank and Computational Exponents: Subadditivity of the border rank feeds directly into upper bounds for asymptotic tensor rank, relevant for bounding the computational complexity of matrix and graph-based multiplication (Christandl et al., 2020).

Example	Additive Bound	Semi-Additive Value	$\delta$
(3,3,3)	68	65	3
(2,2,7)	33	29	4
($n=10$)	2000	$\leq$1301	699
($n_1=n_2=n_3=4$)	141	126	15

6. Limitations and Open Problems

The validity of higher semi-additivity (i.e., $q$-scalability and exact linearity in small-copy data) is not universal. The linearity assumption is stringent, requiring internal consistency conditions (such as reality of discriminants, and monotonicity). Not all resource quantifiers or tensor invariants necessarily satisfy this framework. Analytically, the companion-matrix formalism generalizes to arbitrary $q$, but practical solution of the resulting systems is feasible mainly for $q\leq 4$. Numerically, higher semi-additive formulas empirically fit well even outside the set of exact parameters, suggesting possible avenues for relaxation to approximate $q$-additivity. Open questions include:

• Deriving $q$-additivity properties directly from operational or axiomatic definitions of resource monotones,
• Extending the approach to mixed or hierarchical small-copy inputs,
• Constructing new computable monotones by enforcing $q$-additivity by design (Melo et al., 2022).

7. Implications for Computational Complexity and Future Directions

Demonstrating strict subadditivity of border rank for all orders $k \geq 4$ establishes that naive compositional scaling fails in broad tensor families, with immediate consequence for the asymptotic analysis of computation-related tensors (including those representing generalized matrix multiplication). Although current upper bounds derived from semi-additivity do not yet surpass those obtained via laser or group-theoretic techniques, they are elementary and structurally robust (Christandl et al., 2020). The conceptual expansion via higher semi-additivity offers a powerful analytic tool for both the fundamental understanding and practical calculation of resource scaling in quantum information, tensor theory, and related computational domains.