Maximally Entangled State

Maximally entangled state (or Bell state generalization, $\Phi_d$) is a multi-qubit or multi-level state with maximum entanglement entropy between subsystems. For $n$ qubits, maximally entangled states achieve the upper bound on entanglement for their dimension, representing perfect correlation despite perfect mixedness from single-subsystem perspective.

The $n$-qubit maximally entangled state is:

$$|\Phi_d\rangle = \frac{1}{\sqrt{d}} \sum_{x=0}^{d-1} |x\rangle_A \otimes |x\rangle_B$$

where each subsystem has dimension $d$. For qubits, $d = 2^n$.

The four two-qubit Bell states are the maximally entangled states for a pair of qubits. Each Bell state violates the CHSH inequality maximally and has entanglement entropy $\log_2(2) = 1$ ebit (entanglement bit).

For $n$ qubits, maximally entangled states include:

• Equal superposition of all $2^n$ basis states with all amplitudes equal in magnitude
• Global phase may vary across basis states (as in GHZ states or W states)
• Entanglement entropy $= n$ ebits for a pure maximally entangled state

A bipartite state is maximally entangled if its Schmidt decomposition has:

• Maximum number of non-zero Schmidt coefficients (Schmidt rank = $\min(d_A, d_B)$ for subsystems A and B)
• All Schmidt coefficients equal in magnitude: $\lambda_i = 1/\sqrt{d}$ for all $i$

This ensures subsystem A (or B) alone yields maximum mixedness.

• High entanglement entropy: measured by von Neumann entropy of reduced density matrix
• Violation of separability: non-factorable; Bell inequalities are violated maximally
• Uniform measurement statistics: measuring either subsystem yields all outcomes with equal probability
• Determinism from measurement: knowing outcome of one subsystem fully determines the other (for some states, up to local unitary)

Maximally entangled states may not be unique for a given subsystem structure—the Bell states are one family; others are obtained via local unitaries.

• Quantum teleportation: Bell state (maximally entangled pair) enables teleportation of unknown qubit
• Superdense coding: Bell pair allows two classical bits encoded in one qubit manipulation
• Quantum cryptography: maximum entanglement enables secure key distribution (E91 protocol)
• Quantum channels: Choi-Jamiolkowski correspondence represents channels as maximally entangled states in channel dimension
• Resource theory: quantified by entanglement entropy; consumption in protocols determines advantage

Create maximally entangled states via:

1. Prepare initial product state (e.g., $|0\rangle^{\otimes n}$)
2. Apply Hadamard gates to create superposition
3. Apply entangling gates (CNOT, CZ) to couple subsystems
4. Adjust global phases to match desired state (via single-qubit rotations)

For two-qubit case: Hadamard on first qubit, then CNOT with first as control, produces Φ⁺ Bell state.

Maximally entangled states are realizable on:

• Photonic systems: post-selected generation of Bell states
• Trapped ions: high-fidelity CNOT gates create Bell pairs
• Superconducting qubits: two-qubit gates prepare entanglement
• Neutral atoms: Rydberg interactions enable multi-qubit entanglement

• Entanglement swapping: combining two-qubit maximally entangled states yields weaker entanglement (no $n$-qubit maximum scaling directly)
• Decoherence: maximally entangled states are fragile; loss of coherence in any subsystem breaks maximal entanglement
• Trade-off: increasing $n$ requires more two-qubit gates, each with errors; final fidelity degrades exponentially