Entanglement

A quantum correlation between particles where the state of one instantly relates to the state of another, regardless of distance.

Entanglement is one of the most profound features of quantum mechanics. When particles become entangled, their quantum states become correlated in ways that have no classical explanation. Einstein famously called it “spooky action at a distance.”

The Basic Idea

Consider two qubits in this state (a Bell state):

$$|{\Phi }^{+}⟩=\frac{1}{\sqrt{2}}(|00⟩+|11⟩)$$

This state says: “Both qubits are in superposition, but they’re perfectly correlated.”

• Measure the first qubit as 0 → the second qubit is definitely 0
• Measure the first qubit as 1 → the second qubit is definitely 1

The correlation is instant and works regardless of how far apart the qubits are separated.

What Entanglement Is NOT

• NOT faster-than-light communication: You can’t send information using entanglement alone. The measurement results are random, and you can’t control what you get.
• NOT the same as classical correlation: If I put one red ball and one blue ball in separate boxes and ship them apart, opening one tells me the other’s color. But quantum entanglement is different. Bell’s theorem proves this mathematically.

Creating Entanglement

The standard way to entangle two qubits:

1. Start with $|00⟩$
2. Apply Hadamard to the first qubit: $\frac{1}{\sqrt{2}}(|0⟩+|1⟩)|0⟩$
3. Apply CNOT with first qubit as control: $\frac{1}{\sqrt{2}}(|00⟩+|11⟩)$

Types of Entangled States

Bell States (2 qubits):

• $|{\Phi }^{+}⟩=\frac{1}{\sqrt{2}}(|00⟩+|11⟩)$
• $|{\Phi }^{-}⟩=\frac{1}{\sqrt{2}}(|00⟩-|11⟩)$
• $|{\Psi }^{+}⟩=\frac{1}{\sqrt{2}}(|01⟩+|10⟩)$
• $|{\Psi }^{-}⟩=\frac{1}{\sqrt{2}}(|01⟩-|10⟩)$

GHZ State (3+ qubits): $$|GHZ⟩=\frac{1}{\sqrt{2}}(|000⟩+|111⟩)$$

W State (3 qubits): $$|W⟩=\frac{1}{\sqrt{3}}(|001⟩+|010⟩+|100⟩)$$

Why It Matters

Entanglement is a resource for quantum computing and communication:

Measuring Entanglement

How do you know if a state is entangled? A pure state of two systems is entangled if it cannot be written as a product state $|{\psi }_A⟩\otimes |{\psi }_B⟩$. For mixed states, various entanglement measures exist (concurrence, negativity, entanglement entropy).

See also: Bell State, Bell Inequality, Quantum Teleportation, CNOT Gate