Quantum Entanglement
Quantum entanglement is a phenomenon in which two or more particles become correlated in such a way that the quantum state of each particle cannot be described independently of the others, even when separated by large distances. When a measurement is performed on one entangled particle, the outcome instantaneously determines the corresponding property of its partner, regardless of the distance between them. Einstein famously called this "spooky action at a distance," but experimental tests of Bell's inequalities have confirmed it as a real feature of nature, now foundational to quantum cryptography and quantum teleportation.
Key Formula
|Φ+⟩ = (1/√2)(|00⟩ + |11⟩)
LaTeX: |\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)
| Symbol | Meaning | Unit |
|---|---|---|
| |Φ+⟩ | Bell state — one of four maximally entangled two-qubit states | dimensionless |
| |00⟩ | Both particles in state 0 | dimensionless |
| |11⟩ | Both particles in state 1 | dimensionless |
| 1/√2 | Normalization constant ensuring total probability = 1 | dimensionless |
The Four Bell States (Maximally Entangled Two-Qubit States)
| Bell State | Notation | Expression | Correlation Type |
|---|---|---|---|
| Phi-plus | |Φ+⟩ | (1/√2)(|00⟩ + |11⟩) | Correlated (same outcomes) |
| Phi-minus | |Φ−⟩ | (1/√2)(|00⟩ − |11⟩) | Correlated (same outcomes) |
| Psi-plus | |Ψ+⟩ | (1/√2)(|01⟩ + |10⟩) | Anti-correlated (opposite outcomes) |
| Psi-minus | |Ψ−⟩ | (1/√2)(|01⟩ − |10⟩) | Anti-correlated (opposite outcomes) |
Interactive Tools
Brilliant.org — Quantum Entanglement
In-depth interactive treatment of entanglement and Bell's theorem.
Open Tool
Wikimedia Commons, CC BY-SA
Related Terms
Quantum Superposition
Quantum superposition is the principle that a quantum system can exist in multiple distinct states simultaneously until a measurement is performed, at which point the wave function collapses to a single definite state. Mathematically, the state of a particle is described by a linear combination (superposition) of basis states, each with a complex amplitude whose squared modulus gives the probability of that outcome. The principle underpins interference phenomena, quantum computing (qubits), and famous thought experiments such as Schrödinger's cat.
Quantum Tunneling
Quantum tunneling is the quantum mechanical phenomenon by which a particle penetrates through a potential energy barrier that it classically could not surmount. Unlike classical mechanics, where a particle must have enough energy to overcome a barrier, quantum mechanics allows a non-zero probability of the particle's wave function existing on the other side of the barrier. This effect is responsible for nuclear fusion in stars, the operation of tunnel diodes, scanning tunneling microscopes, and radioactive alpha decay.
Pauli Exclusion Principle
The Pauli Exclusion Principle states that no two identical fermions (particles with half-integer spin) can simultaneously occupy the same quantum state within a quantum system. This principle, formulated by Wolfgang Pauli in 1925, explains the structure of the periodic table and the stability of matter — electrons in an atom must each have a unique set of quantum numbers (n, l, m_l, m_s). It underlies the existence of distinct atomic orbitals, the hardness of solids, and the phenomenon of electron degeneracy pressure in white dwarf stars.
The term "entanglement" was introduced by Erwin Schrödinger in 1935 in his paper responding to the Einstein–Podolsky–Rosen paradox; the German word he used was Verschränkung, meaning "interlocking" or "entwining." The concept of non-local correlations was first articulated by Einstein, Podolsky, and Rosen in their 1935 paper.