What if an eavesdropper Eve measures each bit in either the standard basis or the

Hadamard basis to obtain a bit string z and forwards the measured qubits to Bob. On

average, how many bits of Alice and Bob$$s key does she know for sure after listening in

on the public classical? If Alice and Bob compare s bit values of their strings x

$$

and y

$$

$,$

how likely are they to detect Eve$$s presence?