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Quantum Fidelity between two density operators, $\hat{\rho}$ and $\hat{\sigma}$, is given by $F(\hat{\rho},\hat{\sigma})=\left(Tr\sqrt{\sqrt{\hat{\rho}}\hat{\sigma}\sqrt{\hat{\rho}}}\right)^2$, where $Tr$ represents the trace.

If both the density operators represent pure states: $\hat{\rho}=|\psi\rangle\langle\psi|$ and $\hat{\rho}=|\phi\rangle\langle\phi|$ then this becomes $|\langle\psi|\phi\rangle|^2$.

$|\langle\psi|\phi\rangle|^2 = h\int W_{\psi}(x,p)W_{\phi}(x,p)dxdp$ in terms of the Winger functions (From Eqn. (19) in Case, W. B. (2008). Wigner functions and Weyl transforms for pedestrians. American Journal of Physics, 76(10), 937-946.)

Do we have a similar expression in terms of the Quasi-Probability Distributions: P and Q instead of the density operators when the states are pure?

Is there also a general expression for the case of mixed states?

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