Entanglement

Entanglement is a property of quantum systems in which the quantum state of two or more objects cannot be described independently of each other, even when the objects are spatially separated. Measurements on one part of an entangled system affect the state of the other part in a way that produces correlations beyond those allowed by classical statistics, while not enabling faster-than-light communication.

The concept emerged from the Einstein-Podolsky-Rosen paradox in 1935, which questioned the completeness of quantum mechanics. In 1964, John Bell showed that certain predictions of quantum mechanics for entangled systems would violate inequalities that any local hidden-variable theory would satisfy. Experiments testing Bell inequalities have repeatedly demonstrated results consistent with quantum predictions, strengthening the view that entanglement reflects a fundamental aspect of nature, though early tests left some loopholes that later experiments have aimed to close.

Formally, a pure entangled state cannot be written as a product of states of its subsystems. The Schmidt decomposition provides a useful representation, and entanglement can be quantified by measures such as entanglement entropy. Entanglement can be observed in pairs or larger multiparty systems and is verified through violations of Bell inequalities or other entanglement witnesses.

Entanglement is created and maintained in various platforms, including spontaneous parametric down-conversion in optics, trapped ions, and superconducting qubits. It is sensitive to decoherence and experimental imperfections. Applications include quantum teleportation, superdense coding, and quantum key distribution, as well as serving as a key resource in many quantum information processing tasks.