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Entanglement ​

Entanglement is a uniquely quantum phenomenon where two or more qubits become correlated in ways impossible classically. Measuring one instantly determines the other, regardless of distance.

What is Entanglement? ​

When qubits are entangled, their states cannot be described independently. The classic example is a Bell state:

(|00> + |11>) / sqrt(2)

In this state:

  • Each qubit individually has 50% chance of being 0 or 1
  • But they always measure to the same value
  • If one measures 0, the other is guaranteed to be 0
  • If one measures 1, the other is guaranteed to be 1

The qubits share a single quantum state that cannot be factored into independent parts.

Creating Bell States ​

Kettle provides the bell() function to create an entangled pair:

How Bell States Work ​

You can create a Bell state manually with Hadamard and CNOT:

The circuit works like this:

  1. hadamard puts the first qubit in superposition: (|0> + |1>)/sqrt(2)
  2. cnot correlates the second qubit with the first
  3. Result: (|00> + |11>)/sqrt(2) - entangled!

Verifying Entanglement ​

The signature of entanglement is perfect correlation:

Every pair matches - this is impossible to achieve classically without pre-shared information.

Multi-Qubit Entanglement ​

Entanglement extends beyond two qubits. A GHZ state entangles three or more qubits:

Entanglement is Not Cloning ​

Entanglement might seem like copying, but it is fundamentally different:

Key differences from cloning:

  • Cloning would create an independent copy
  • Entanglement creates correlated qubits
  • Entangled qubits share one quantum state, not two copies

The no-cloning theorem says you cannot copy an unknown quantum state. Kettle enforces this with linear types.

Applications ​

Entanglement enables:

  • Quantum teleportation - Transfer quantum states using entanglement + classical bits
  • Superdense coding - Send 2 classical bits using pre-shared entanglement + 1 qubit
  • Quantum cryptography - Detect eavesdroppers via entanglement
  • Quantum algorithms - Speed up computation via entangled parallelism

SWAP Test ​

The SWAP test uses entanglement to measure how similar two quantum states are:

This is a practical example of using entanglement for quantum state comparison.