# 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?

Things state very clear in your link, I've just quoted the main thing and added some reasonings.

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 $$F_{*}(\rho, \tau)=F_{*}(\hat{\rho}, \hat{\tau})=(\operatorname{Tr}|\sqrt{\hat{\rho}} \sqrt{\hat{\tau}}|)^{2}\tag{2}$$

So from eq.(2), we can see that it's obviously bounded by 0 and 1.