1/2, $|T_{\rm in}\rangle$ goes from $|+\rangle$ to $|-\rangle$. The (unnormalized) output states that may occur for this target qubit are $$\begin{aligned}
|T_\pm\rangle &\equiv& C|+\rangle \pm \frac{S}{\sqrt{2}}|-\rangle \\ %[.05in]
|T_E\rangle &\equiv& \frac{S}{\sqrt{2}}|-\rangle.\end{aligned}$$

Here is how the FPB probe works. When Alice uses the $H/V$ basis for her photon transmission, Eve’s CNOT gate effects the following transformation, $$\begin{aligned}
|H\rangle|T_{\rm in}\rangle &\longrightarrow&
|H\rangle|T_-\rangle + |V\rangle|T_E\rangle \label{Hin_out} \\ %[.05in]
|V\rangle|T_{\rm in}\rangle &\longrightarrow&
|V\rangle|T_+\rangle +|H\rangle|T_E\rangle,\label{Vin_out} %\\[-.075in] \nonumber\end{aligned}$$ where the kets on the left-hand side denote the Alice$\otimes$Eve state of the control and target qubits at the CNOT’s input and the kets on the right-hand side denote the Bob$\otimes$Eve state of the control and target qubits at the CNOT’s output. Similarly, when Alice uses the $\pm 45^\circ$ basis, Eve’s CNOT gate has the following behavior, $$\begin{aligned}
|\mbox{$+$}45^\circ\rangle|T_{\rm in}\rangle &\longrightarrow&
|\mbox{$+$}45^\circ\rangle|T_+\rangle + |\mbox{$-$}45^\circ\rangle|T_E\rangle 
\label{plus_in_out}\\ %[.05in]
|\mbox{$-$}45^\circ\rangle|T_{\rm in}\rangle &\longrightarrow&
|\mbox{$-$}45^\circ\rangle|T_-\rangle +|\mbox{$+$}45^\circ\rangle|T_E\rangle.
\label{minus_in_out}\end{aligned}$$ Suppose that Bob measures in the basis that Alice has employed *and his outcome matches what Alice sent. Then Eve can learn their shared bit value, once Bob discloses his measurement basis, by distinguishing between the $|T_+\rangle$ and $|T_-\rangle$ output states for her target qubit. Of course, this knowledge comes at a cost: Eve has caused 