In quantum mechanics, entanglement swapping is a protocol to transfer quantum entanglement from one pair of particles to another, even if the second pair of particles have never interacted. This process may have application in quantum communication networks and quantum computing.

Entanglement swapping has two pairs of entangled particles: (A, B) and (C, D). Pair of particles (A, B) is initially entangled, as is the pair (C, D). The pair (B, C) taken from the original pairs, is projected onto one of the four possible Bell states, a process called a Bell state measurement. The unmeasured pair of particles (A, D) can become entangled. This effect happens without any previous direct interaction between particles A and D.

Entanglement swapping is a form of quantum teleportation. In quantum teleportation, the unknown state of a particle can be sent from one location to another using the combination of a quantum and classical channel. The unknown state is projected by Alice onto a Bell state and the result is communicated to Bob through the classical channel. In entanglement swapping, the state from one of the two sources is the quantum channel of teleportation and the state from the other source is the unknown being sent to Bob.

The mathematical expression for the swapping process is:

$${\displaystyle \left|\psi \right\rangle _{AB}\otimes \left|\psi \right\rangle _{CD}\xrightarrow {{\text{BSM}}_{BC}} \left|\psi \right\rangle _{AD}}$$

In this expression, ${\displaystyle \left|\psi \right\rangle _{XY}}$ refers to an entangled state of pair of particles (X,Y) while BSM indicates Bell state measurement. A Bell state is one of four specific states of representing two particles with maximal entanglement; a Bell state measurement projects a quantum state onto this basis set.

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

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 may support safely transferring quantum information over long routes.