AEP in Multipartite Entanglement

• The asymptotic equipartition property (AEP) is a principle that converts one-shot entanglement measures into additive, entropy-based quantifiers in large-copy regimes.
• It employs smoothing and regularization to mitigate non-asymptotic fluctuations, resulting in convex combinations of bipartite entanglement entropies.
• AEP underpins LOCCq transformations by establishing robust rate conversion metrics, ensuring operational consistency in multipartite quantum systems.

The asymptotic equipartition property (AEP) in the context of multipartite entanglement measures provides a rigorous bridge between general, potentially irregular one-shot quantifiers of entanglement and the unique, additive, and operationally meaningful entropy-based measures governing large-block (asymptotic) regimes. In multipartite systems, where entanglement structure is more complex and less canonical than in bipartite settings, AEP yields a canonical regularization procedure for subadditive entanglement measures, leading to convex combinations of bipartite entanglement entropies. This formalism plays a pivotal role in understanding and characterizing resource conversion rates under operationally relevant scenarios such as LOCC with sublinear quantum communication ($\mathrm{LOCC}_q$).

1. Mathematical Formulation of the Multipartite Entanglement AEP

Consider a pure state %%%%1%%%% on a multipartite Hilbert space. Let $E$ be an entanglement monotone defined for pure states. The “smoothed” version of $E$ is given by

$$E^\varepsilon(\psi) \coloneqq \inf_{\varphi: |\langle\varphi|\psi\rangle|^2 \ge 1 - \varepsilon} E(\varphi),$$

where smoothing is over all nearby pure states $\varphi$ within Bures (fidelity) distance $\varepsilon$. The regularized (asymptotic) functional, denoted $\Phi(E)$, is then defined as

$$\Phi(E)(\psi) \coloneqq \lim_{\varepsilon \to 0} \limsup_{n \to \infty} \frac{1}{n} E^\varepsilon(\psi^{\otimes n}).$$

This mirrors classical AEP: the limit quantifies the “typical” value of $E$ for large numbers of independently prepared copies, after smoothing out non-asymptotic fluctuations.

2. Structural Conditions for the AEP

The class of entanglement measures to which this regularization applies is characterized by the following weak structural requirements:

• Subadditivity: $E(\psi_1 \otimes \psi_2) \le E(\psi_1) + E(\psi_2)$ for all pure $\psi_1, \psi_2$.
• Direct-sum logarithmic boundedness: $$E(\psi_1 \oplus \cdots \oplus \psi_\ell) \le \max_{i} E(\psi_i) + \log \ell$$.
• Monotonicity or monotonicity on average under LOCC.

These requirements ensure that regularization yields a measure which is weakly additive and asymptotically continuous in the sense that for large $n$ and small $\varepsilon$, the per-copy rate is stable.

3. Operational Significance in $\mathrm{LOCC}_q$ Transformations

The principal motivation for introducing AEP in the entanglement context arises from the scenario of many-copy $\mathrm{LOCC}_q$ conversions—transformations under local operations and classical communication supplemented with a sublinear quantity of quantum communication. In such settings, questions of optimal conversion rates between multipartite pure states reduce asymptotically to comparisons between their regularized entanglement rates.

In the bipartite case, the von Neumann entropy of the reduced state $H(\mathrm{Tr}_A|\psi\rangle\langle\psi|)$ uniquely determines the asymptotic (rate) behavior, such as the distillable entanglement and cost under LOCC. The multipartite AEP formalism elevates this additive, “ergodic” structure to general multipartite entanglement measures, ensuring that, after regularization, operational rates depend only on convex combinations of subsystem entropies.

4. Regularization, Smoothing, and Functional Properties

The procedure combines smoothing (minimizing over an $\varepsilon$-neighborhood in fidelity or trace distance) and regularization (taking the $n \to \infty$ limit, normalized by $1/n$) to wash out single-copy idiosyncrasies and irregularities. The regularization map $\Phi$ is idempotent and restricts to functionals that are subadditive, weakly additive (the $n$-copy value scales linearly), and asymptotically continuous. In effect, for any map $E$ in the admissible class $\mathcal{F}_{\mathrm{sub},k}$, the output $\Phi(E)$ is a well-behaved asymptotic monotone suited for resource-theoretic interpretations.

5. Application to Rényi-type Multipartite Entanglement Measures

The evaluation of this program on explicit Rényi-type multipartite entanglement measures is instructive. Consider $E^{(\alpha,\theta)}$, a family of measures parameterized by $\alpha \in [0,1)$ and a probability vector $\theta = (\theta_1, ..., \theta_k)$ over subsystem partitions. Bounds satisfied are: $$E^{(1, \theta)}(\psi) = \sum_{j} \theta_j H(\mathrm{Tr}_j \psi),\qquad E^{(0, \theta)}(\psi) \le \sum_{j} \theta_j \log\operatorname{rk}(\mathrm{Tr}_j \psi),$$ where $H(\cdot)$ is the von Neumann entropy and $\operatorname{rk}(\cdot)$ the rank. Upon regularization and smoothing, the limit

$$\lim_{\varepsilon \to 0} \limsup_{n \to \infty} \frac{1}{n} \inf_{\varphi\text{ close to } \psi^{\otimes n}} E^{(\alpha,\theta)}(\varphi) = \sum_{j} \theta_j H(\mathrm{Tr}_j \psi)$$

holds for all $\alpha \in [0,1)$, indicating that the choice of $\alpha$ is entirely washed out in the asymptotic regime. That is, all such regularized Rényi-type measures collapse to convex combinations of subsystem entropies.

6. Asymptotic Measure as Convex Combination of Bipartite Entanglement Entropies

The core outcome is that, in the multipartite setting, under the above procedural limits, the regularized asymptotic measure always takes the form

$$H^{\theta}(\psi) = \sum_j \theta_j H(\mathrm{Tr}_j \psi).$$

Thus, the multipartite AEP asserts that the only “robust” asymptotic quantifiers of pure state entanglement are convex linear combinations of the bipartite (reduced state) entropies, with the weights $\theta_j$ reflecting possible operational choices or symmetries.

This aligns precisely with the classical AEP: the regularized smooth Rényi entropies all collapse asymptotically to the Shannon entropy. The multipartite extension thus rigorously underpins the practice of quantifying asymptotic multipartite entanglement resources using additive, entropy-based functionals.

7. Summary Table: Smoothing and Regularization for Multipartite Entanglement Measures

Step	Definition / Formula	Resulting Property
Smoothing	$E^\varepsilon(\psi)$	Stabilizes non-asymptotic fluctuations
Regularization	$$\Phi(E)(\psi) = \lim_{\varepsilon \to 0} \limsup_{n\to\infty} \frac{1}{n} E^\varepsilon(\psi^{\otimes n})$$	Yields weakly additive, continuous monotone
Application to $E^{(\alpha,\theta)}$	$$\Phi(E^{(\alpha,\theta)})(\psi) = \sum_{j} \theta_j H(\mathrm{Tr}_j \psi)$$	Collapses to convex sum of subsystem entropies

8. Significance and Implications

The result establishes a multipartite AEP: for any continuous, subadditive, monotone pure-state multipartite entanglement measure, the process of smoothing and regularization recovers a well-defined asymptotic measure that is necessarily a convex sum of bipartite entanglement entropies. This justifies the universality and operational primacy of the von Neumann entropy in the extensive regime and demonstrates that more exotic, one-shot multipartite measures do not introduce new asymptotic features beyond what is already captured by convex combinations of subsystem entropies. This is a critical insight for the structure theory of quantum entanglement and for resource-theoretic analyses in quantum information (Bugár, 26 Sep 2025).