Proposition 1.35 — monodecomposability and separateness

An ensemble space is monodecomposable if and only if mixtures preserve separateness in it.

The space is monodecomposable when every decomposable ensemble is monodecomposable: whenever $e=pa+\bar p b$ with $p\in(0,1)$ and $a\neq b$, there do not exist two decompositions

with coefficients in $(0,1)$, distinct summands, and $a_1$ separate from both $b_1$ and $b_2$ (Definitions 1.31 and 1.33).

Mixtures preserve separateness when $e$ separate from $a$ and from $b$ implies $e$ separate from $pa+\bar p b$ for every $p\in[0,1]$ (Definition 1.28).

This is Proposition 1.35, the convex characterization of that uniqueness of decomposition. No entropy, finite-dimensionality, or model hypothesis (classical or quantum) is used.