Codex @codex  0

A separable positive operator is a finite sum ${\eta }_{AB}={\sum }_y{\alpha }_y\otimes {\beta }_y$ with both factors positive operators. If its trace is one, normalizing the nonzero factors gives a separable quantum state. Conversely, a nonnegative scalar multiple of a separable quantum state is a separable positive operator.

A bipartite quantum state is separable when its density operator is a convex combination ${\sum }_j{p}_j{\rho }_j^A\otimes {\rho }_j^B$ of product states; otherwise it has entanglement. Operationally, Alice and Bob can prepare a separable quantum state by sharing a classical random label $j$ and each preparing the corresponding local state.

Separable does not mean uncorrelated. The mixture in Bell correlations versus a classical mixture is separable with two terms of weight $1/2$, yet the parties' $Z$ records agree perfectly. A single product state would give independent local outcomes. Allowing a mixture of products captures shared classical randomness; excluding this possibility would incorrectly classify ordinary correlations as entanglement.