# Quantum Entanglement Swapping

Quantum Entanglement Swapping is an essential idea in quantum mechanics. It involves entanglement from one pair of particles to another, even if those new particles have never interacted before. This process is very important for building quantum communication networks, enabling quantum teleportation and advancing quantum computing.

## History

The idea of quantum entanglement swapping came from physicists Marek Żukowski, Anton Zeilinger, Michael A. Horne, and Artur K. Ekert in 1993. Their important paper in Physical Review Letters introduced that one can extend entanglement from one particle pair to another using a method called Bell state measurement.

### Key historical milestones

* 1993: The concept of entanglement swapping was first proposed by Żukowski and his team.
* 1998: Jian-Wei Pan and his group conducted the first experiment on entanglement swapping. They used entangled photons to show successful transfer of entanglement between pairs that never interacted.
* 2000s: Later experiments took this further, making it work over longer distances and with more complex quantum states.
* 2017: Demonstrated in satellite-based experiments, allowing entanglement distribution over 1200 kilometers.

## Concept
### Basic principles:

Quantum entanglement swapping has three pairs of entangled particles: (A, B), (C, D), & (E, F). Particles A & B are initially entangled, just like C & D. By applying a process called Bell state measurement to one particle from each pair (like B and C), the unmeasured particles (A and D) can become entangled. This happens without any direct interaction between them.
The measurement collapses the states of B and C into one of four Bell states. Due to the laws of quantum mechanics, this instantly determines the state of A and D.

### Mathematical representation:

In quantum mechanics, we can use a Bell state to represent two particles in an entangled system. The mathematical expression for the swapping process looks like this:
$\left|\psi \right\rangle _{AB}\otimes \left|\psi \right\rangle _{CD}\xrightarrow {{\text{BSM}}_{BC}} \left|\psi \right\rangle _{AD}$
In this expression, $\left|\psi \right\rangle _{XY}$ refers to the state of X & Y particles while BSM indicates Bell state measurement.

## Development and expansions
### Quantum repeaters and long-distance communication:

One main use of quantum entanglement swapping is for creating quantum repeaters. These devices help stretch out quantum communication networks by allowing entanglement to be shared over long regions. Performing entanglement swapping at certain points acts like relaying information without loss.

### Multi-particle entanglement:

Scientists are pushing these ideas further into multi-particle setups! Now they're discovering ways to create complex entangled states known as GHZ states (Greenberger-Horne-Zeilinger states). These states are super important for quantum error correction and making fault-tolerant quantum computers.

### Satellite-based quantum communication:

Experiments on satellite-based quantum communication showed how entanglement can link ground stations via satellites while using entanglement swapping to increase range. This marks a huge leap toward building a global quantum internet.

## Applications
### Quantum teleportation:

Entanglement swapping plays an essential role in quantum teleportation, where the state of a particle can be sent from one spot to another without moving the particle itself. This relies on using entangled pairs through the swapping process.

### Quantum cryptography:

In the field of quantum cryptography, it helps secure communication channels better. By utilizing swapped entangtlements between particles' pairs, it is possible to generate secure encryption keys that should be protected against eavesdropping.

### Quantum networks:

Quantum Entanglement Swapping also serves as a core technology for designing quantum networks, where many nodes-like quantum computers or communication points-link through these special connections made by entangled links. These networks support safely transferring quantum information over long routes and contribute significantly to building the emerging quantum internet.