# Properties of the generalized fidelity for subnormalized states

The generalized fidelity for quantum states that may be sub-normalized is given by (Defn 3.12)

$$F_{*}(\rho, \tau):=\left(\operatorname{Tr}|\sqrt{\rho} \sqrt{\tau}|+\sqrt{(1-\operatorname{Tr} \rho)(1-\operatorname{Tr} \tau)}\right)^{2},$$

where $$|A| = \sqrt{A^\dagger A}$$.

Is this generalized fidelity also bounded between $$0$$ and $$1$$? How can one see this?

Uhlmann's theorem (Theorem 3.10) adapted to the generalized fidelity states that \begin{aligned} F_{*}(\rho, \tau) &=\max _{\varphi, \vartheta} F_{*}(\varphi, \vartheta)=\max _{\vartheta} F_{*}(\phi, \vartheta), \quad \text { where } \\ \sqrt{F_{*}(\varphi, \vartheta)} &=|\langle\varphi \mid \vartheta\rangle|+\sqrt{(1-\operatorname{Tr} \varphi)(1-\operatorname{Tr} \vartheta)} \end{aligned}\tag{1} and $$\varphi$$ and $$\vartheta$$ range over all purifications of $$\rho$$ and $$\tau$$, respectively, and $$\phi$$ is a fixed purification of $$\rho$$. Moreover, using the operators $$\hat{\rho}$$ and $$\hat{\tau}$$ defined in the preceding section, we can write $$F_{*}(\rho, \tau)=F_{*}(\hat{\rho}, \hat{\tau})=(\operatorname{Tr}|\sqrt{\hat{\rho}} \sqrt{\hat{\tau}}|)^{2}\tag{2}$$