Quantum Entanglement
Quantum entanglement is a physical phenomenon that occurs when a group of particles is generated, interact, or share spatial proximity in a way such that the quantum state of each particle cannot be described independently of the state of the others, including when the particles are separated by a large distance.1
Historical Development
The paradox of action at a distance was discussed by Albert Einstein and others in 1935. Einstein and others argued that the absence of such a mechanism in quantum mechanics was an indication that the theory was incomplete. However, John Stewart Bell later formulated Bell's theorem, which showed that no local hidden variable theory could reproduce the predictions of quantum mechanics.3
"I like to think that the quantum mechanics is a very beautiful theory, but I am not sure it is the whole truth."
— Erwin Schrödinger, 1935
Mathematical Formulation
In quantum mechanics, an entangled state of a composite system is a state that cannot be written as a product of states of its individual subsystems. For two qubits, a maximally entangled state is given by:
\|\Phi^+\rangle = \frac{1}{\sqrt{2}} (\|00\rangle + \|11\rangle)
Applications
- Quantum Computing: Entanglement is a key resource for quantum speedup in algorithms like Shor's and Grover's.
- Quantum Cryptography: Protocols like E91 use entanglement to guarantee secure key distribution.
- Quantum Teleportation: Transfers quantum states between particles using classical communication and shared entanglement.
Research continues into macroscopic entanglement and its potential implications for our understanding of spacetime and gravity.7