Maximally Entangled States in Quantum Systems

• Maximally entangled states are quantum states where subsystems exhibit complete uncertainty, ensuring maximal entanglement across balanced bipartitions.
• They underpin key quantum tasks such as teleportation, quantum error correction, and secret sharing through structures like Bell states, AME states, and MES.
• Their construction relies on advanced tools from graph theory, combinatorial designs, and coding theory, providing benchmarks for entanglement measures and LOCC conversions.

A maximally entangled state is a pure or mixed quantum state that exhibits the highest achievable entanglement according to specific structural, operational, or measurement criteria. In the context of bipartite systems, the archetypal maximally entangled state is the Bell state for two qubits, which achieves maximal entanglement for all bipartitions. For multipartite systems, several notions and classifications arise—including absolutely maximally entangled (AME) states, maximally multipartite entangled states (MMES), extremal maximally entangled (EME) states, and maximally entangled sets (MES)—each tailored to reflect the intricate structure of multipartite entanglement and the operational tasks relevant to quantum information science.

1. Operational Definitions and Entanglement Measures

The defining property of a maximally entangled state is that certain reduced density matrices for chosen subsystems are maximally mixed. For a pure n-partite state $|\psi\rangle$, the AME condition is $$\mathrm{Tr}_{A^c}(|\psi\rangle\langle\psi|) = \frac{\mathbb{I}}{d^{|A|}}, \qquad \forall A,\, |A| \le \left\lfloor n/2 \right\rfloor,$$ where $|A|$ specifies a subset of parties and $d$ is the local dimension. This guarantees maximal von Neumann entropy $S(\rho_A) = |A| \log d$ for all reductions up to half the system's size, reflecting maximal uncertainty and entanglement across bipartitions.

Several quantitative entanglement measures are used:

• Tangle ($\tau$ and $k$-tangle): Originally introduced for three qubits as the 3-tangle (Wootters), the tangle generalizes to four qubits (4-tangle), capturing genuine multipartite residual entanglement not decomposable into lower-order correlations. The 4-tangle has an operational meaning as the residual left after subtracting all possible bipartite entanglements; explicitly, $4\tau_1 - 3\tau_2 = \tau_{ABCD}$, with $\tau_1$ and $\tau_2$ averages over 1-vs-3 and 2-vs-2 partitions, respectively (Gour et al., 2010).
• Entropy measures: Linear entropy, Rényi $\alpha$-entropy, and Tsallis $\alpha$-entropy of entanglement are used to quantify mixedness of reductions. For a pure state $\vert\psi^{AB}\rangle$ and reduced density matrix $\rho_r$, the Rényi entropy is

$$E_R^{(\alpha)}(\vert\psi^{AB}\rangle) = \frac{1}{1-\alpha} \log \operatorname{Tr}(\rho_r^\alpha),$$

with similar expressions for Tsallis and linear entropy. These enable fine characterization of entanglement structure across different cuts.

• Concurrence and I-concurrence: The square of the concurrence for a bipartition $X|X'$ is $E^2_{X|X'} = 2[1 - \operatorname{tr} (\rho_X^2)]$. Maximally entangled states minimize $\operatorname{tr} (\rho_X^2)$ for all partitions (Bag et al., 26 Jan 2025).

2. AME States, Existence, and Structure

Absolutely maximally entangled (AME) states are pure states that are maximally entangled across all bipartitions up to half the system size. For qubits ($d=2$), AME states are known to exist only for $n=2,3,5,6$ (Helwig et al., 2013, Zhang et al., 19 Nov 2024, Huber et al., 2016). Their defining property is every reduction to $\lfloor n/2 \rfloor$ parties is maximally mixed. An AME($n,d$) state can be constructed via classical coding theory—for example, from maximum distance separable (MDS) codes or orthogonal arrays (Helwig et al., 2013, Goyeneche et al., 2015).

Beyond the existence conjectures and explicit constructions, AME states are deeply tied to combinatorial designs (e.g., mutually orthogonal Latin squares, frequency squares, biunimodular vectors) and multi-unitary matrices, wherein the coefficient tensor is unitary under any balanced matrix reshaping (Goyeneche et al., 2015, Casas et al., 7 Apr 2025, Rajchel-Mieldzioć et al., 6 Aug 2025).

When AME states are not possible (e.g., $n=4$ qubits (Huber et al., 26 Jun 2025)), the problem becomes to maximize the number of maximally mixed reductions of size $\lfloor n/2 \rfloor$. This is formalized in the "quantum extremal number" $Qex(n)$, the largest possible number of maximally mixed half-system reductions in any $n$-qubit pure state (Zhang et al., 19 Nov 2024).

3. Families of Maximally Entangled States: Structure and Uniqueness

For four qubits, the set of "maximally entangled states" (in the sense of maximizing average bipartite tangle) forms a four-parameter family, characterized by

$$\mathcal{M} = \Big\{ \sum_{j=0}^3 \sqrt{p_j} e^{i\theta_j} u_j \;|\; \sum p_j = 1,\; \sum p_j e^{2i\theta_j} = 0 \Big\},$$

with $u_j$ an orthogonal basis (typically Bell-like), and the zero 4-tangle condition enforcing genuine four-party entanglement (Gour et al., 2010). However, most states in this family are not simultaneous optimizers for other entropic measures.

Remarkably, only two unique four-qubit states (up to local unitaries) maximize the average Tsallis or Rényi $\alpha$-entropy: the state $|L\rangle$ for $\alpha \ge 2$ and $|M\rangle$ for $0 < \alpha < 2$ (Gour et al., 2010). Three specific cluster states realize the maximal average Rényi entropy for $\alpha \ge 2$, each achieving 2 ebits in two of three $2|2$ bipartitions and 1 ebit in the third.

For multipartite systems, the Maximally Entangled Set (MES) comprises states from which all other fully entangled states outside MES can be reached via LOCC, but none in the set can be reached from outside (Vicente et al., 2013). In four-qubit systems, almost all states are isolated (no deterministic LOCC conversions), with a measure-zero subset being convertible; for three qubits, the MES has measure zero.

4. Graph Theory, Coding, and Extremal Constructions

The combinatorial challenge of maximizing the count of maximally mixed reductions is mapped to extremal hypergraph theory via Turán-type problems (Zhang et al., 19 Nov 2024). For an $n$-qubit state, every maximally mixed $k$-subset corresponds to a $k$-edge in a $k$-uniform hypergraph. A key result is that for $n=4$, $Qex(4)=4$, for $n=7$, $Qex(7)=32$, and for $n=8$, $Qex(8)=56$. Construction methods include:

• Explicit graph states: For each half-size subset $K \subset [n]$, the submatrix $A_{K \times K^c}$ of the adjacency matrix must be full rank for the reduction to be maximally mixed (Zhang et al., 19 Nov 2024).
• Probabilistic methods: Computing the probability that a random graph yields a maximally mixed reduction allows analytic lower bounds on $Qex(n)$.
• Coding-theoretic and invariant theory proofs: For example, the nonexistence of a four-qubit AME state is proven via polynomial invariants and Rains' shadow inequalities, showing that the required local invariants cannot be satisfied consistently (Huber et al., 26 Jun 2025).

5. Applications in Quantum Information Processing

Maximally entangled (especially AME) states are indispensable in quantum information tasks:

• Quantum secret sharing and error correction: AME($2m,d$) states implement threshold QSS schemes and enable construction of pure quantum error-correcting codes with maximal distance (Helwig et al., 2013, Goyeneche et al., 2015, Rajchel-Mieldzioć et al., 6 Aug 2025).
• Teleportation and entanglement swapping: AME states generalize parallel and open-destination teleportation protocols and facilitate multipartite entanglement swapping, underpinning quantum repeater networks and fault-tolerant quantum computation (Helwig et al., 2013, Casas et al., 7 Apr 2025, Rajchel-Mieldzioć et al., 6 Aug 2025).
• Device-independent certification: In certain pseudo-telepathy games (e.g., weak projection games), only strategies employing maximally entangled states can lead to perfect correlations, providing a robust method for device-independent certification of entanglement (Mančinska, 2015).
• Multipartite cryptography and network protocols: The property that any subset up to half of the parties is maximally mixed enables information-theoretic security in secret sharing, distributed consensus, and voting protocols (Helwig et al., 2013, Rajchel-Mieldzioć et al., 6 Aug 2025).

6. Open Problems and Future Directions

Several central problems remain open:

• Existence questions: For many combinations of $n$ (number of parties) and $d$ (local dimension), it is unknown whether AME($n,d$) states exist, or what is the maximal number of maximally mixed reductions ($Qex(n)$) achievable (Zhang et al., 19 Nov 2024, Rajchel-Mieldzioć et al., 6 Aug 2025). Notably, no seven-qubit pure AME state exists (Huber et al., 2016).
• Classification and invariants: The structure and classification of AME states (local unitary and SLOCC classes) become increasingly rich for higher $N$ and $d$. For certain cases, continuously many LU classes exist (Rajchel-Mieldzioć et al., 6 Aug 2025). The computation of higher-order invariants, such as hyperdeterminants, remains challenging (Goyeneche et al., 2015).
• Robustness to noise: The entanglement properties of AME states are robust under some noise channels (e.g., depolarizing channels retain symmetry), but can depend on choice of local unitary representatives under other channels (such as dephasing), leading to symmetry breaking in entanglement distribution (Stawska et al., 10 May 2025).
• Non-stabilizer and higher-dimensional constructions: While many AME states are stabilizer or graph states, new constructions via non-stabilizer protocols and high-dimensional qudits (using multi-unitary gates and combinatorial designs) are under active development (Casas et al., 7 Apr 2025, Rajchel-Mieldzioć et al., 6 Aug 2025).
• Mixed maximally entangled states (MME): The multipartite generalization of maximally entangled mixed states poses a stringent requirement: every pure component in every convex decomposition must itself be maximally entangled. These states set absolute benchmarks for entanglement measures and remote state preparation (Hedemann, 2021).

7. Mathematical Structures Underpinning Maximally Entangled States

The construction and characterization of maximally entangled states connect with deep mathematical concepts:

• Combinatorial designs: Orthogonal Latin squares, orthogonal frequency squares, MDS codes, and orthogonal arrays offer systematic ways to construct AME states, especially for higher dimensions (Goyeneche et al., 2015, Rajchel-Mieldzioć et al., 6 Aug 2025).
• Multi-unitarity: AME states of $2k$ parties require $k$-unitary matrices—tensors unitary over every balanced reshaping—ensuring maximality of entanglement for all bipartitions (Goyeneche et al., 2015, Casas et al., 7 Apr 2025).
• Invariant and operator theory: LU and SLOCC equivalence, as well as degree-4 polynomial invariants, shape the classification landscape for multipartite entanglement (Huber et al., 26 Jun 2025).
• Extremal graph theory: Turán-type results provide upper bounds on $Qex(n)$, linking the combinatorics of entanglement to classical extremal graph (and hypergraph) theory (Zhang et al., 19 Nov 2024).

In combination, these mathematical frameworks reveal both the richness and the limitations of maximal entanglement in quantum systems, drive the discovery of new entangled resources, and anchor the application of these states to foundational and practical tasks in quantum information theory.