Corrigendum: A limit formula for the quantum fidelity (2013 J. Phys. A: Math. Theor. 46 025304)

On page 5, there is an error in the proof of theorem 1. This theorem is still valid but the actual proof is as follows.

Proof. Specify the definition (14) to the case where $\sigma =\left| \varphi \right\rangle \left\langle \varphi \right|$, i.e.,

$$\begin{eqnarray}&&{{C}_{s}}={\rm Tr}\left( {{\rho }^{s}}\left| \varphi \right\rangle {{\left\langle \varphi \right|}^{1-s}} \right)\;.\end{eqnarray} \tag{ 22 }$$

For every $s\in (0,1)$ we can use the property of the projector

$$\begin{eqnarray}&&\left| \varphi \right\rangle {{\left\langle \varphi \right|}^{1-s}}=\left| \varphi \right\rangle \left\langle \varphi \right|,\end{eqnarray} \tag{ 23 }$$

and write

$$\begin{eqnarray}&&{{C}_{s}}=\left\langle \varphi \right|{{\rho }^{s}}\left| \varphi \right\rangle \;.\end{eqnarray} \tag{ 24 }$$

Now, we can always decompose ρ as

$$\begin{eqnarray}&&\rho =\mathop{\sum }\limits_{k}p_{k}^{1/2}\left| k \right\rangle \left\langle k \right|\;,\end{eqnarray} \tag{ 25 }$$

where ${{p}_{k}}\in [0,1]$ for any k, and $\{\left| k \right\rangle \}$ is an orthonormal set

$$\begin{eqnarray}&&\langle k|k^{\prime} \rangle ={{\delta }_{kk^{\prime} }}.\end{eqnarray} \tag{ 26 }$$

Taking the s-power of (25), we get

$$\begin{eqnarray}&&{{\rho }^{s}}=\mathop{\sum }\limits_{k}p_{k}^{s/2}\left| k \right\rangle \left\langle k \right|,\end{eqnarray} \tag{ 27 }$$

for any $s\in (0,1)$. Using the latter expression in (24), we achieve

$$\begin{eqnarray}&&{{C}_{s}}=\mathop{\sum }\limits_{k}p_{k}^{s/2}{{\left| \left\langle k \right|\left. \varphi \right\rangle \right|}^{2}}.\end{eqnarray} \tag{ 28 }$$

Finally, taking the limit of $s\to {{1}^{-}}$, we derive

$$\begin{eqnarray}&&\mathop{{\rm lim} }\limits_{s\to {{1}^{-}}}{{C}_{s}}=\mathop{\sum }\limits_{k}p_{k}^{1/2}{{\left| \left\langle k \right|\left. \varphi \right\rangle \right|}^{2}}\end{eqnarray} \tag{ 29 }$$

$$\begin{eqnarray}&&=\left\langle \varphi \right|\rho \left| \varphi \right\rangle =F(\rho ,\left| \varphi \right\rangle ),\end{eqnarray} \tag{ 30 }$$

which corresponds to the result of equation (21). □

Correspondingly, the proof of the corollary on page 6 must also be modified. It must be replaced by the following:

Proof. This is a trivial consequence of the previous theorem. In fact, we have that C_s in equation (28) is manifestly non-increasing in s. As a consequence, we have

$$\begin{eqnarray}&&C=\mathop{{\rm inf} }\limits_{s\in (0,1)}{{C}_{s}}=\mathop{{\rm lim} }\limits_{s\to {{1}^{-}}}{{C}_{s}}=F(\rho ,\left| \varphi \right\rangle ),\end{eqnarray} \tag{ 33 }$$

which completes the proof. □