Subadditive Entanglement Measures

Updated 29 September 2025

Subadditive entanglement measures are defined by the inequality E(ρ ⊗ σ) ≤ E(ρ) + E(σ) and quantify quantum correlations in both bipartite and multipartite systems.
They encompass entropic, geometric, and operational methods that guide practical protocols like state distillation, dilution, and LOCC transformations.
Subadditivity establishes polygon and triangle relations, setting structural constraints and ensuring continuity important for quantum resource theories.

Subadditive entanglement measures are central to the characterization and quantification of quantum correlations in multipartite and bipartite quantum systems. These measures, defined by the property that the entanglement of a composite state does not exceed the sum of entanglements of its constituents, embody a broad class of functionals including entropic, geometric, and operationally-motivated quantities. Their subadditivity leads to profound structural constraints on the distribution and conversion of entanglement, defines the geometric landscape of multipartite entanglement, and determines operational performance in protocols such as distillation, dilution, and state transformation under local operations and classical communication (LOCC).

1. Definition and Core Properties

A function $E$ defined on quantum states is called subadditive if, for any states $\rho$ and $\sigma$ ,

$E(\rho \otimes \sigma) \leq E(\rho) + E(\sigma).$

Strict additivity corresponds to equality, but subadditivity only requires the weaker inequality. Subadditivity manifests in both entropic measures (e.g., von Neumann, Rényi, and Tsallis entropies of reduced density matrices) and in measures extended via convex roof constructions or operational definitions such as the (regularized) relative entropy of entanglement, geometric measure, or computational analogues.

Subadditivity is foundational for the following reasons:

It codifies the intuition that "entanglement cannot be freely created" by tensor product composition.
It supports the structure of constraints such as polygon (triangle) relations and monogamy inequalities.
It underpins continuity and asymptotic equipartition properties (Bugár, 26 Sep 2025).

2. Subadditivity and Geometric/Entropic Measures

Many canonical entanglement measures are subadditive by construction:

Entropic Measures:

For the von Neumann entropy $S(\rho)$ and generalizations including the Rényi $S_\alpha(\rho)$ and Tsallis $T_q(\rho)$ entropies, subadditivity for bipartite states takes the form

$S(\rho_{AB}) \leq S(\rho_A) + S(\rho_B).$

This extends to multipartite systems and is tightly connected with polygon (and triangle) relations (Liu et al., 4 Jan 2024).

Geometric Measure of Entanglement (GME):

Defined via the maximal squared overlap with product states:

$\Lambda^2(\psi) = \max_{|\phi\rangle \in \text{PROD}} |\langle \phi | \psi \rangle|^2,$

with $G(\psi) = -2\log \Lambda(\psi)$ . For "non-negative" states (states with real, non-negative entries in a product basis), GME is strictly additive (Zhu et al., 2010). However, for antisymmetric or generic multipartite states, GME can be strictly subadditive, i.e., $G(\rho \otimes \sigma) < G(\rho) + G(\sigma)$ (Dilley et al., 29 Mar 2025).

Informationally Complete Measures:

Measures constructed from all moments or the full spectrum of the reduced density matrix are fine-grained and can exhibit subadditive properties. However, they also enable distinctions that are inaccessible to coarser, simply subadditive measures (Jin et al., 2022, Wang et al., 8 May 2024).

3. Polygon and Triangle Relations: Geometric Structure

Polygon and triangle relations geometrically express subadditivity across multipartitions:

Polygon Relation:

For multipartite pure states, with $E(\rho_{A_i})$ the entropy of the reduced state of party $A_i$ :

$E(\rho_{A_i}) \leq \sum_{j \neq i} E(\rho_{A_j}).$

The set of entanglement entropies $\{E(\rho_{A_1}), \ldots, E(\rho_{A_N})\}$ forms the sides of a geometric polygon, and the relation expresses that each individual entanglement does not exceed the sum of the rest (Liu et al., 4 Jan 2024).

Triangle Relation (Tripartite):

For any subadditive bipartite measure $\mathcal{E}$ and $\alpha \in [0,1]$ ,

$\mathcal{E}^\alpha_{i|jk} \leq \mathcal{E}^\alpha_{j|ik} + \mathcal{E}^\alpha_{k|ij}$

for all permutations of $\{A,B,C\}$ . The triangle area, defined via Heron's formula, serves as a faithful quantifier of genuine tripartite entanglement, with area strictly positive iff the state is genuinely entangled (Ge et al., 2023).

Equivalence to Subadditivity:

The polygon relation holds for all states if and only if the underlying entropic measure is subadditive (Liu et al., 4 Jan 2024). For pure states, the triangle relation (for $\alpha \leq 1$ ) is equivalent to subadditivity of the bipartite measure.

Relation Type	Inequality Form	Validity
Polygon	$E(\rho_{A_i}) \leq \sum_{j \neq i} E(\rho_{A_j})$	iff $E$ is subadditive
Triangle	$\mathcal{E}^\alpha_{i\|jk} \leq \mathcal{E}^\alpha_{j\|ik} + \mathcal{E}^\alpha_{k\|ij}$	$\forall$ subadditive $E$ and $\alpha \in [0,1]$

4. Subadditivity in Operational and Computational Measures

Subadditivity also arises in operational and computational contexts:

Relative Entropy of Entanglement (REE):

For REE, $E_R(\rho \otimes \sigma) \leq E_R(\rho) + E_R(\sigma)$ follows from the variational definition. Non-additivity for certain multipartite states has been established (Zhu et al., 2010), and the asymptotic equipartition property for regularized, additive versions leads to convex combinations of marginal entropies (Bugár, 26 Sep 2025).

Computational Entanglement Measures:

In computational settings (e.g., measures constrained by efficient LOCC), only lower/upper bound forms of subadditivity and superadditivity are generally satisfied:

$\hat{E}_D(\rho_1 \otimes \rho_2) \geq \hat{E}_D(\rho_1) + \hat{E}_D(\rho_2) \quad \text{(superadditive lower bound)},$

$\hat{E}_C(\rho_1 \otimes \rho_2) \leq \hat{E}_C(\rho_1) + \hat{E}_C(\rho_2) \quad \text{(subadditive upper bound)},$

where $\hat{E}_D$ and $\hat{E}_C$ are computational distillable entanglement and cost, respectively (Ryzov et al., 26 Sep 2025).

Detection and Hypothesis-Testing Measures:

Certain measures quantifying distinguishability from separable sets—based on "reverse" quantum relative entropy or sandwiched Rényi divergence—are additive and thus subadditive for families such as maximally correlated states (Hayashi et al., 2023).

5. Continuity, Asymptotics, and Regularization

The continuity of subadditive measures is fundamentally limited:

Asymptotic Continuity:

Any non-constant, additive or subadditive entanglement measure (invariant under local unitaries) cannot have better than asymptotic continuity, i.e.,

for suitable $K$ and system dimension $d$ (Coladangelo et al., 2019).

Asymptotic Equipartition Property (AEP):

For subadditive multipartite measures satisfying mild conditions, the regularization and smoothing operation (infimum over $\epsilon$ -balls, normalization across $n$ copies, and $\lim_{\epsilon\to0} \limsup_{n\to\infty}$ ) reduces any such measure to a weakly additive, asymptotically continuous form, typically a convex combination of marginal von Neumann entropies (Bugár, 26 Sep 2025).

Reduction to Bipartite Entanglement Entropies:

The AEP implies that for pure multipartite states, all regularized, smooth subadditive measures collapse to weighted sums of bipartite entanglement entropies, mirroring the classical Shannon paradigm.

6. Interplay with Monogamy, LOCC, and Resource Theory

The subadditivity of entanglement measures enters into foundational resource-theoretic and structural results:

Monogamy vs. Subadditivity:

Subadditivity leads to polygon/triangle inequalities (upper bounds), whereas monogamy inequalities provide lower bounds on the shareability of entanglement. For convex-roof measures (such as squared concurrence), monogamy often holds in qubit systems (Yang et al., 2020), but for entropic measures (like Rényi or Tsallis), the polygon relation is a subadditivity-induced upper bound (Liu et al., 4 Jan 2024).

LOCC Transformations:

Subadditivity (and more generally, additivity and monotonicity) governs the reversibility and rates of asymptotic state transformation under LOCC and LOCC $_q$ scenarios. The failure of additivity, for example in relative entropy of entanglement, obstructs reversible interconversion of generic multipartite states via standard resources (e.g., GHZ states) (Zhu et al., 2010, Bugár, 26 Sep 2025).

Practical Quantification:

Experimentally feasible subadditive measures, such as moment-based entanglement monotones (Wang et al., 8 May 2024) and informationally complete constructs (Jin et al., 2022), retain subadditivity in the presence of noise and partial information, making them robust tools for computational and laboratory platforms.

7. Non-Additive, Superadditive, and Interpolating Monotones

Constructs exist that interpolate between additive, subadditive, and even superadditive regimes:

Weighted Operator Geometric Means:

The use of weighted operator geometric means enables new constructions of subadditive and submultiplicative monotones between flattening ranks and generalized Rényi blocks (Bugár et al., 2022).

Superadditivity and Lower Bounds:

For every monotone constructed from entropic or geometric data, one can identify pointwise smaller additive/multiplicative monotones and construct superadditive lower bounds, capturing finer gradations in multipartite entanglement (Bugár et al., 2022).

Measure Type	Subadditivity	Additivity	Superadditivity
Standard entropic	$E(\rho \otimes \sigma) \leq E(\rho) + E(\sigma)$	$=$ when strict (rare)	Lower bounds occasionally
Polytope/geometric	Triangle / Polygon	$=$ for non-negative states	Possible lower bounds
"Interpolated" monotone	Between flattening rank and $R$ block	Can interpolate	New monotones via geometric means

References

Additivity and non-additivity of multipartite entanglement measures (Zhu et al., 2010)
Moments based entanglement criteria and measures (Wang et al., 8 May 2024)
Faithful geometric measures for genuine tripartite entanglement (Ge et al., 2023)
The polygon relation and subadditivity of entropic measures for discrete and continuous multipartite entanglement (Liu et al., 4 Jan 2024)
Asymptotic equipartition property of subadditive multipartite entanglement measures on pure states (Bugár, 26 Sep 2025)
Exploring Non-Multiplicativity in the Geometric Measure of Entanglement (Dilley et al., 29 Mar 2025)
Entanglement measures for detectability (Hayashi et al., 2023)
Informationally complete measures of quantum entanglement (Jin et al., 2022)
Interpolating between Rényi entanglement entropies for arbitrary bipartitions via operator geometric means (Bugár et al., 2022)
Properties of computational entanglement measures (Ryzov et al., 26 Sep 2025)
Additive entanglemement measures cannot be more than asymptotically continuous (Coladangelo et al., 2019)

These works collectively establish the centrality of subadditivity in entanglement theory, underpinning geometric structure, operational performance, and the ultimate reduction of multipartite quantification to entropic balances across subsystem marginals.