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The fidelity between two density matrix $\rho$ and $\sigma$ is the following:

$$F(\rho,\sigma)=\operatorname{Tr}\left[\sqrt{\sqrt{\rho} \sigma \sqrt{\rho}} \right]^2$$

If one of the two state is a pure state the expression is very easy, but in general it is this one. To compute it I would naïvely diagonalise $\rho$ to find $\sqrt{\rho}$, then compute $\sqrt{\rho} \sigma \sqrt{\rho}$, then diagonalize again to find the square root: $ \sqrt{\sqrt{\rho} \sigma \sqrt{\rho}}$, take the trace and square.

It is a "complicated" thing to do.

I wondered if there are tricks to compute fidelity between two mixed states ? Also, are there some "usual" tricks to make proof based on that (like proving inequalities).

Again my "confusion" is that this formula for two mixed state is not easy to handle so I wondered if I see it under the wrong angle ?

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    $\begingroup$ you can compute it as the sum of the square roots of the eigenvalues of $\rho\sigma$, which just so happens to be always diagonalisable, see this question. I don't know if that qualifies as a "trick". $\endgroup$
    – glS
    Aug 15 '20 at 11:44

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