Separable state

Remarks

Example: on $\C^2 \otimes \C^2$, the state $\sigma = \tfrac12 (|00\rangle\langle 00| + |11\rangle\langle 11|)$ is separable. Indeed $\sigma = \tfrac12 |0\rangle\langle 0| \otimes |0\rangle\langle 0| + \tfrac12 |1\rangle\langle 1| \otimes |1\rangle\langle 1|$, with weights $\tfrac12, \tfrac12$. The maximally mixed state $\tfrac14 I = \tfrac12 I \otimes \tfrac12 I$ is separable with a single term.

Non-example: let $\psi = (|00\rangle + |11\rangle)/\sqrt{2}$. A pure state $|\psi\rangle\langle\psi|$ is an extreme point of the set of states, so in any decomposition as above every term with $p_i > 0$ equals $|\psi\rangle\langle\psi|$. Then $|\psi\rangle\langle\psi|$ would be a product $\rho^A \otimes \rho^B$, which forces $\psi$ to be a product vector, and it is not one.

Equivalently, the separable states form the convex hull of the pure product states $|a\rangle\langle a| \otimes |b\rangle\langle b|$. By Carathéodory's theorem, at most $(d_A d_B)^2$ terms are needed, where $d_A = \dim H_A$ and $d_B = \dim H_B$. In infinite dimension, one takes the closure of this convex hull in the trace norm.