Maximal Entanglement Limit (MEL)

• Maximal Entanglement Limit (MEL) is a fundamental concept defining the upper bound on quantum entanglement achievable in a system, measured via metrics like von Neumann and Rényi entropies.
• MEL is attained through protocols including dual-unitary quantum circuits, quantum walks, and Absolutely Maximally Entangled (AME) states that ensure optimal entropy distribution across subsystems.
• MEL has practical implications in quantum error correction, secret sharing, and scattering processes by guiding the design of optimal quantum resource states and operational protocols.

The Maximal Entanglement Limit (MEL) is a multifaceted concept denoting the algebraic or operational upper bounds on achievable quantum entanglement within a given system or transformation. MEL provides a theoretical ceiling for entanglement entropy, quantifiable by different measures—ranging from bipartite concurrence and von Neumann entropy to multipartite invariants such as multi-unitarity, and extends to constraints governing quantum statistical ensembles and dynamical protocols. MEL is central to quantum information theory, quantum many-body physics, statistical mechanics, and quantum field theory, as it delineates the nonclassical constraints imposed by unitarity, dimensionality, and symmetry on the possible quantum correlations in composite systems.

1. Formal Definitions and Entropic Bounds

The basic setting for MEL is a pure state $|\Psi\rangle$ in a composite Hilbert space $\mathcal{H}_A \otimes \mathcal{H}_B$, with the reduced density operator on subsystem $A$ given by $$\rho_A = \operatorname{Tr}_B |\Psi\rangle\langle\Psi|$$. The maximal value of the von Neumann entropy, $S(\rho_A) = -\operatorname{Tr}[\rho_A \log \rho_A]$, is achieved when $\rho_A$ is maximally mixed, i.e., $\rho_A = I_{d_A}/d_A$, yielding $S_\text{max} = \ln d_A$ for subsystem dimension $d_A$ (Kharzeev, 1 Jan 2026, Huang, 2021). At this point, all outcomes in the Schmidt basis are equally likely, corresponding to a "Page-typical" state for $d_B \gg d_A$.

In the multipartite context, the MEL is realized by Absolutely Maximally Entangled (AME) states. For an $n$-partite pure state $|\Psi\rangle \in (\mathbb{C}^d)^{\otimes n}$, AME requires that all reduced density matrices on subsets of up to $k = \lfloor n/2 \rfloor$ parties are maximally mixed: $$\rho_A = \operatorname{Tr}_{A^c} |\Psi\rangle\langle\Psi| = \frac{I_{d^{|A|}}}{d^{|A|}}\,, \qquad S(\rho_A) = |A| \log d\,,$$ with $A$ any subset of size $|A| \leq k$ (Helwig et al., 2012, Goyeneche et al., 2015, Zhang et al., 2024).

For continuous variables and non-Gaussian operations, MEL is expressed via Rényi-$2$ entanglement: for any bipartite pure state $|\psi\rangle$, $$\mathcal{E}_R(|\psi\rangle) = -\ln \operatorname{Tr}[\rho_A^2]$$, and single-photon subtraction/addition cannot increase $\mathcal{E}_R$ by more than $\ln 2$ (one ebit), regardless of mode number or purity (Zhang et al., 2021).

2. Dynamical and Operational Mechanisms for Realizing MEL

Quantum Circuits and Dual Unitarity: In locally interacting quantum circuits, entanglement entropy grows at a velocity $v_E$ up to an upper bound set by the Lieb–Robinson light-cone ($v_E \leq 1$ for $q$-dimensional sites). Achieving $v_E=1$ requires that the two-site gate $U$ be "dual-unitary," i.e., unitary under exchange of time and space directions:

• Unitarity: $U U^\dagger = I$
• Dual unitarity: $(UF)^\Gamma (UF)^{\Gamma\dagger} = I$, with $F$ the swap and $\Gamma$ partial transpose

Exactly dual-unitary gates (e.g., SWAP, Fourier, self-dual kicked Ising) ensure that each circuit layer enables the maximum entropy flow, with the entanglement profile showing a "zigzag" pattern that propagates unchanged at maximum possible slope (Zhou et al., 2022).

Quantum Walks: One-dimensional quantum walks with either position-inhomogeneous coins (Zhang et al., 2022) or dynamically disordered coins (1305.4191) asymptotically generate maximal bipartite entanglement between coin and walker. In the random-coined case, the coin state approaches the maximally mixed state ($S_E\to1$) for any initial condition, while inhomogeneity enables exact saturation at every odd step and rapid approach on even steps.

Spin Chains and Ancillae: In open quantum systems with ancilla and spin chains, the multipartite entanglement loss (MEL) between the chain and ancilla is defined as the difference between the total spin collective fluctuations and the spin chain's quantum Fisher information. When all multipartite spin entanglement is lost to the ancilla, MEL saturates at the total spin variance, scaling as $O(L)$, where $L$ is the system size (Szabo et al., 2021).

3. Multiparty Structures and Combinatorial Limits

AME States and Multi-Unitarity: For $N$-qudit states, AME implies each $k$-party reduction ($k \leq N/2$) is maximally mixed, and the global coefficient tensor $t_{i_1\cdots i_N}$ becomes $k$-unitary (unitary on all respective splits). Existence depends on $N$ and $d$: it is known for specific $(N,d)$—e.g., AME(6,2) ("hexabit") and AME(4,3) ("tetratrit")—but prohibited for $N=4,8$ and $d=2$ (Goyeneche et al., 2015). The connection to classical MDS codes provides both explicit constructions and necessary conditions (Goyeneche et al., 2015, Helwig et al., 2012).

Maximal Entanglement in Absence of AME: For $n$-qubit systems where AME$(n,2)$ does not exist ($n\notin\{2,3,5,6\}$), the quantum extremal number $Qex(n)$ gives the maximal number of $\lfloor n/2 \rfloor$-qubit subsystems that can be maximally mixed in any state. $Qex(n)$ is bounded above by Turán-type extremal combinatorics and below by explicit (e.g., graph-state) constructions. E.g., $Qex(8) = 56$ (Zhang et al., 2024).

Multipartite Entanglement Measures: The GME-AME measure $E(|\Psi\rangle)$ smoothly interpolates between $0$ (biseparable) and $1$ (AME), reflecting the maximal entanglement possible for an $n$-party $d$-level system. $E=1$ if and only if the state is AME (V et al., 2024).

4. MEL in Quantum Dynamics and Statistical Ensembles

Entanglement Growth and Page's Theorem: For a Haar-typical pure state in $\mathcal{H}_A\otimes\mathcal{H}_B$ with $d_B \gg d_A$, Page's theorem gives the average entropy

$$\langle S_A \rangle = \ln d_A - \frac{d_A}{2 d_B} + O(1/d_B^2)$$

so that for large systems, nearly maximal entropy is generic (Kharzeev, 1 Jan 2026). Dynamical approaches (global quenches, random circuits) quickly drive subsystems toward this limit under generic, nonintegrable Hamiltonians.

Limits Due to Conservation Laws: Under local or all-to-all Hamiltonian evolution, starting from a product state, the entanglement entropy is always bounded a finite amount below the maximal value, by $\Delta \sim n/N$ for local Hamiltonians and $\Delta \sim n^2/N^2$ (or $n^4/N^4$) for all-to-all (SYK-like) interactions, reflecting the restriction to symmetry sectors (Huang, 2021).

Statistical and High-Energy Physics: In high-energy collisions, the final state particle multiplicity spectrum near $z=n/\langle n\rangle=2$ approaches a purely exponential KNO scaling function $\Psi(z)\sim e^{-z}$, corresponding to maximal (Shannon/von Neumann) entropy under mean constraints. This statistical MEL reflects saturation of the optical theorem and maximal final-state entanglement (Ouchen et al., 23 Nov 2025). In quantum field theory, tracing over unobservable degrees of freedom (e.g., light-cone time in QCD) produces reduced density matrices indistinguishable from thermal ensembles, making probabilistic partonic interpretations a direct consequence of quantum entanglement (Kharzeev, 1 Jan 2026).

5. Physical and Foundational Implications

Quantum Information Tasks and Error Correction: AME states, by attaining the MEL, are optimal for threshold quantum secret sharing and multipartite teleportation. Quantum error-correcting codes that saturate the Singleton bound correspond to AME states in the appropriate dimension (Helwig et al., 2012, Goyeneche et al., 2015).

Fundamental Interactions and MaxEnt Principle: Demanding that scattering amplitudes allow outgoing states to saturate the MEL constrains the structure of fundamental vertices. E.g., in $2\rightarrow2$ tree-level QED scattering, the requirement of maximal concurrence $\Delta=1$ at $\theta=\pi/2$ recovers the standard gauge-invariant QED vertex and, for $Z$ boson processes, predicts $\sin^2\theta_W=1/4$ (Cervera-Lierta et al., 2017, Cervera-Lierta, 2019).

Topological Quantum Phases: Localized Majorana zero-modes at the edges of finite-size Kitaev tubes realize MEL between distant edges, producing Bell states whose entanglement is rigorously one ebit, entirely independent of system size—an emergent quantum resource for robust, nonlocal qubit operations (Wang et al., 2017).

6. Representative Table: MEL in Different Contexts

Context	Criterion for MEL	Maximal Entanglement Value
Bipartite finite systems	$\rho_A = I/d_A$	$S_\text{max} = \ln d_A$
Pure $n$-qudit AME state	All $k\le n/2$ reductions maximal	$S(\rho_A)=k \log d$
2-mode CV, single-photon subtraction	$\mathcal{E}_R \le \ln 2$	1 ebit (Rényi-2)
Coin-walker QRWs	$\rho_\text{coin}\to I/2$	$S_E \to 1$ (qubit)
QED tree-level scattering	Concurrence $\Delta=1$	Bell state pair
High-energy pp KNO scaling	$S=\ln\langle n\rangle+1$	$z=2$ point, $\Psi(z)=e^{-2}$
Many-body random state (Page)	Trace-distance to $I/d_A$	$S_A \simeq \ln d_A$

7. Open Questions and Outlook

The maximal entanglement limit exposes foundational and practical frontiers. Explicit AME states are unknown or provably nonexistent for many $(n, d)$ (Zhang et al., 2024, Goyeneche et al., 2015). The extremal combinatorial problem for $Qex(n)$ remains unsolved for large $n$. In quantum thermodynamics, the precise conditions and timescales for saturation of the MEL under various Hamiltonian classes are open to further mathematical refinement (Huang, 2021, Kharzeev, 1 Jan 2026). Operationalizations of MEL—as in resource theories, cryptography, and topologically protected systems—remain a frontier for both experimental verification and conceptual development.

The unifying theme across domains is that the maximal entanglement limit marks a fundamental boundary set by quantum theory: a boundary realized—sometimes exactly, sometimes only approximately—by optimal protocols, topological orders, or statistical ensembles, and which often defines the ultimate resources for both computation and physical law.