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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 $\rho$.

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Hint: One method to prove HUP for mixed states from HUP for pure states is to use the purification trick.

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  • $\begingroup$ 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? $\endgroup$ – K. Sadri Apr 1 '18 at 15:04
  • $\begingroup$ By the way, I didn't make any specific use of purification. The same result could've been achieved by taking expected values any way. So I guess I'm missing something $\endgroup$ – K. Sadri Apr 1 '18 at 15:10

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