Figure 1

Geometric representation of the key-controlled \(\vert \Psi \rangle \), \(\vert R \rangle \), and \(\vert L \rangle \) in the Bloch sphere. The red arrow represents the secret key \(\hat{k}_{AB} = ( \pi /4,0 )\) and the red sphere represents a specific \(\vert \Psi \rangle \) among all possible points represented by the orange solid line. The blue and cyan spheres represent \(\vert R \rangle \) and \(\vert L \rangle \) generated by the rotation operator \(\mathcal{R}_{\hat{k}_{AB}}\)