Quantum Cryptography Based on Two Mixed States

Although it is known that any two nonorthogonal pure states can constitute a secure quantum cryptosystem, the generalization of this scheme to the use of two mixed states is not trivial. It is shown here that even if a condition corresponding to the nonorthogonality in the pure-state case is satisfied, the mixed-state cryptosystem is still vulnerable to attack by an eavesdropper. A necessary and sufficient condition for the secure communication is derived. It states that the two mixed states must be connected by a rotation operator with a nonorthogonal angle.