Nonfactorizability

Nonfactorizability is the property of a system or model in which its description cannot be written as a product of independent parts. In probability theory, a joint distribution P(A,B) is factorizable if P(A,B) = P(A)P(B). Nonfactorizable distributions exhibit correlations that cannot be reduced to independent marginals. The concept also applies to functions and physical models where a two-part description cannot be separated into a product of a function of one part and a function of the other.

In quantum physics, nonfactorizability is closely tied to quantum entanglement and the failure of local hidden-variable theories to explain observed correlations. For measurement outcomes a,b of distant subsystems given settings x,y, a local realistic model imposes a factorization of joint probabilities: P(a,b|x,y) = ∑_λ p(λ) P(a|x,λ) P(b|y,λ). Quantum states can produce joint probabilities that do not admit such a factorization, violating Bell-type inequalities. This nonfactorizability is considered a signature of quantum nonlocality, reflecting limitations of classical descriptions rather than signaling faster-than-light communication.

Beyond physics, nonfactorizability appears in mathematics and computation to describe objects that cannot be written as a product of independent components. For example, a function F(x,y) is nonfactorizable if there are no A(x) and B(y) with F(x,y) = A(x)B(y). In data analysis, nonfactorizable models capture interactions that cannot be represented by separate effects. The term is often used in discussions of entanglement, local realism, and the foundational implications of Bell's theorem.

See also:

- Factorization

- Entanglement

- Local realism

- Bell's theorem

- CHSH inequality