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

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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

$$ \rho_{AB} = \sum_i p_i \rho_{i,A}\otimes\sigma_{i,B}. $$

Thus, the partial transpose

$$ \rho_{AB}^{T_B} = \sum_i p_i \rho_{i,A}\otimes\sigma_{i,B}^T $$

is a sum of tensor products of positive operators and therefore is itself positive.

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