**Appendix B**

Using the notation of Appendix A, direct calculations show that, for Γ defined in (12), we have:

$$
\Gamma = p\sqrt{s\_1 + 2\Re(u\_1 u\_2^\* s\_2)} + (1 - p)\sqrt{s\_1 + 2\Re(u\_2 u\_3^\* s\_2)},
\tag{A10}
$$

where ) is the real part of a number, and

$$s\_1 \equiv (p^2 + (1-p)^2) \cdot r\_{4\prime} \tag{A11a}$$

$$s\_2 \equiv p(1-p) \cdot \left( (r\_1 + iq\_1)r\_4 - (r\_2 - iq\_2)^2 \right). \tag{A11b}$$

The states 0 and 1, obtained by conditioning upon the system state in the computational basis and tracing out the second environmental qubit, are equal,

$$\mathbf{a}\_0 = \begin{bmatrix} p & 0 \\ 0 & 1-p \end{bmatrix}'\tag{A12a}$$

$$\varrho\_1 = \begin{bmatrix} 1 + p(2r\_4 - 1) - r\_4 & (1 - 2p)(r\_2 + iq\_2) \\ (1 - 2p)(r\_2 - iq\_2) & p(1 - 2r\_4) + r\_4 \end{bmatrix}.\tag{A12b}$$

From the above, it follows that, for

$$
\mu = -p(1-p)(2r\_4 - 1) + 0.5r\_{4\prime} \tag{A13}
$$

we have

$$\sqrt{\varrho\_0}\varrho\_1\sqrt{\varrho\_0} - \mu\mathbb{1}\_2 = \begin{bmatrix} (p-0.5)r\_4 & \sqrt{p(1-p)}(1-2p)(r\_2+iq\_2) \\ \sqrt{p(1-p)}(1-2p)(r\_2-iq\_2) & -(p-0.5)r\_4 \end{bmatrix} . \tag{A14}$$

Using (A9), we find that the eigenvalues of (A14) are ±*ν*, where

$$\nu = 0.5|1 - 2p|\sqrt{r\_4(r\_4 - 4(p - p^2) \cdot (r\_4 - 1))}.\tag{A15}$$

Thus, we have the following closed form for (13):

$$\mathcal{F}(\varrho\_0, \varrho\_1) = \sqrt{\mu + \nu} + \sqrt{\mu - \nu}. \tag{A16}$$
