an error event whenever Alice and Bob choose a common basis and her target qubit’s output state is $|T_E\rangle$. To maximize the information she derives from this intrusion, Eve applies the minimum error probability receiver for distinguishing between the single-photon polarization states $|T_+\rangle$ and $|T_-\rangle$. This is a projective measurement onto the polarization basis $\{|d_+\rangle,|d_-\rangle\}$, shown in Fig. 3 and given by $$\begin{aligned}
|d_+\rangle &=& \frac{|+\rangle + |-\rangle}{\sqrt{2}} = |0\rangle \\ 
|d_-\rangle &=& \frac{|+\rangle - |-\rangle}{\sqrt{2}} = |1\rangle. \end{aligned}$$*

Two straightforward calculations will now complete our review of the FPB probe. First, we find the error probability that is created by Eve’s presence. Suppose Alice and Bob use the $H/V$ basis and Alice has sent $|H\rangle$. Alice and Bob will incur an error if the control$\otimes$target output from Eve’s CNOT gate is $|V\rangle|T_E\rangle$. The probability that this occurs is $\langle T_E| T_E\rangle = S^2/2 = P_E$. The same conditional error probability ensues for the other three error events, e.g., when Alice and Bob use the $\pm 45^\circ$ basis, Alice sends $|$$+45^\circ\rangle$, and the CNOT output is $|$$-45^\circ\rangle|T_E\rangle$. It follows that the unconditional error probability incurred by Alice and Bob on their sift events is $P_E$.

Now we shall determine the 