# 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.

## Definition for Multiple Qubits

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, $d = 2^n$.

## Two-Qubit Case

The four two-qubit [[quantum-state-bell|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).

## Multi-Qubit Generalization

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 [[quantum-state-ghz|GHZ states]] or [[quantum-state-w|W states]])
- Entanglement entropy $= n$ ebits for a pure maximally entangled state

## Schmidt Decomposition

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.

## Properties

- **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)

## Distinction from GHZ and W States

| State | Entanglement Structure |
|-------|------------------------|
| [[quantum-state-ghz|GHZ]] | All-or-nothing; global phase determines correlation |
| [[quantum-state-w|W]] | Distributed; exactly one excitation |
| Maximally entangled | Maximum entropy in reduced density matrices |

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 Information Applications

- **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

## Construction

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.

## Physical Realization

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

## Limits and Scalability

- **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