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I have a question while reading the quantum fidelity definition in Wikipedia Fidelity of quantum states, at the end of the Definition section of quantum fidelity formula, it says Explicit expression for qubits. If rho and sigma are both qubit states, the fidelity can be computed as: $$ F(\rho, \sigma)=\operatorname{tr}(\rho \sigma)+2 \sqrt{\operatorname{det}(\rho) \operatorname{det}(\sigma)} $$ So I'm really confused about where this formula comes from. I tested several simple examples, it seems that the formula is not suitable to calculate the fidelity between two mixed states while the dimensions of the density matrices are not $2\times2$ (e.g. for the case both of the two matrices are diag(0.5,0,0,0.5), the fidelity calculated by this formula is 0.5 not 1.

However, if one of the density matrices is in pure state, it seems the result now is always correct even though the dimensions of the density matrices are bigger than $2\times2$.

I'm wondering how to prove this formula and is it always safe to use it while one of the density matrices is in pure state...

Thanks in advance!

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The general expression for the fidelity is $$ F(\rho,\sigma)=\left(\text{Tr}\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}\right)^2=(\text{Tr}|\sqrt{\rho\sigma}|)^2. $$ Assume $\rho$ and $\sigma$ are $2\times 2$ matrices. Then $\sqrt{\rho\sigma}$ is also a $2\times 2$ matrix which we assume to have eigenvalues $\lambda_1$ and $\lambda_2$. Thus, $$ F=(|\lambda_1|+|\lambda_2|)^2=\lambda_1^2+\lambda_2^2+2|\lambda_1\lambda_2|. $$

However, we also have that $$ \text{Tr}(\rho\sigma)=\lambda_1^2+\lambda_2^2 $$ and $$ \text{det}(\rho\sigma)=\lambda_1^2\lambda_2^2, $$ so $\sqrt{\text{det}(\rho\sigma)}=|\lambda_1\lambda_2|$. So, that proves the relation specifically for $2\times 2$ matrices. As you stated, it generally does not hold for larger matrices.

In the special case of $\rho$ being pure, it does hold, but it's far easier simply to calculate $F=\langle\psi|\sigma|\psi\rangle=\text{Tr}(\rho\sigma)$.

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  • $\begingroup$ Thank you very much!! btw, should the square of the fidelity (the first formula of the answer F=Tr()^2) here be outside the trace (like (Tr())^2)? and another question that I'm not very clear is, is it always necessary to apply the absolute value to the density operator (Tr(|sqrt(sigma*rho)|)^2)? it seems the absolute value of the operator is calculated as: |A|=sqrt(A^(dagger) A ) as shown in the "Fidelity of quantum states" page of Wikipedia. $\endgroup$ – Yaoling Yang Jun 14 at 9:47
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    $\begingroup$ Yes, the square is applied to the whole trace (otherwise it would remove the square root in some of the expressions). I would advise leaving the absolute value calculating in there (or replacing it with an equivalent expression). It might be that you can introduce additional properties that let you remove it, but I don't know that you can. $\endgroup$ – DaftWullie Jun 14 at 9:54

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