# How can one argue that the partial transpose $\rho^{T_B}$ of a general separable state is positive?

How can one argue that the partial transpose $$\rho^{T_B}$$ of a general separable state is positive?

First observe that if $$R$$ and $$S$$ are positive (aka positive semi-definite) operators and $$q\in [0, +\infty)$$ then $$R^T$$, $$qR$$, $$R\otimes S$$ and $$R+S$$ are all positive operators.
By definition, if $$\rho_{AB}$$ is separable then there exist positive operators $$\rho_{i,A}$$ and $$\sigma_{i,B}$$ with unit trace and $$p_i\in[0, 1]$$ such that
$$\rho_{AB} = \sum_i p_i \rho_{i,A}\otimes\sigma_{i,B}.$$
$$\rho_{AB}^{T_B} = \sum_i p_i \rho_{i,A}\otimes\sigma_{i,B}^T$$