Nonseparability

Nonseparability is the property of a system in which the whole cannot be described merely by the states of its parts. In other words, the state of the whole is not reducible to a simple combination of the states of its subsystems, even when the subsystems are spatially separated or treated independently.

In quantum mechanics, nonseparability is closely tied to entanglement. A composite system with Hilbert space HA ⊗ HB is separable if its state ρ can be written as a sum of product states ρ = ∑i pi (ρA^i ⊗ ρB^i) with pi ≥ 0. If no such convex decomposition exists, the state is entangled and nonseparable. Entangled states exhibit correlations that cannot be explained by local hidden variables, a fact supported by violations of Bell inequalities and by numerous experiments. Schrödinger coined the term nonseparability to describe this holistic feature, where the properties of subsystems are defined only in the context of the whole.

Nonseparability also has implications beyond physics. In philosophy of science, it challenges views that assume complete independence of distant parts and supports more holistic interpretations of reality. In statistics and related fields, a joint distribution is nonseparable if it cannot be factored into independent marginals, reflecting strong or contextual dependence between variables.

Overall, nonseparability highlights fundamental limits to reducing complex systems to independent components and underpins key phenomena and technologies in quantum information science, such as quantum teleportation, superdense coding, and quantum computation.