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I'm studying the measurement in quantum computation. It's known that the trace is related to the expectation value and the probability of getting certain outcomes. However, when the trace is a complex value, how is it related to the probability or the counting number of outcomes in real world?

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$\text{Tr}(AB)$ is always real and non-negative if $A,B$ are positive semi-definite hermitian matrices.

To see this note that $A = UDU^\dagger$, for some unitary $U$ and diagonal matrix $D$ with $d_{ii} \ge 0$.
Then $\text{Tr}(AB) = \text{Tr}(UDU^\dagger B) = \text{Tr}(DU^\dagger B U).$
But $B^\prime = U^\dagger B U$ is also positive semi-definite and hermitian, therefore it has non-negative numbers on diagonal $b^\prime_{ii} \ge 0$. Now $\text{Tr}(DB^\prime) = \sum_i d_{ii}b^\prime_{ii} \ge 0$.

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  • $\begingroup$ Thank you Danylo! $\endgroup$ – raycosine Jul 15 '19 at 5:43

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