# Can we write the Quantum Fidelity between two density operators in terms of Quasi-Probability Distributions: $P$, $Q$ and $W$?

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?