Uncertainty relation in mixed states

If a system is in a pure state $|\psi\rangle\langle\psi|$ we have $$\sigma_A\sigma_B\geq\frac{1}{2}|\langle[A,B]\rangle|.$$ Generalize this and find an uncertainty relation for an arbitrary mixed state $\rho$.

• The most obvious purification I can think of for a mixed state $\rho=\sum_n p_n|n\rangle\langle n|$ is $\psi=\sum_n \sqrt{p_n}|n,n\rangle$ which eventually yields $\sigma_A\sigma_B\geq\frac{1}{2}|\sum_{n,m}\sqrt{p_np_m}\langle n,n|[A,B]|m,m\rangle|$ I don't see any obvious way to proceed. Could you (or someone) say if there is a $neat$ expression at all? – K. Sadri Apr 1 '18 at 15:04