Extended space for quantum cryptography using mixed states

A formulation is presented for quantum cryptography based on two mixed states in rigged Hilbert space. This is distinct from the usual scheme for quantum cryptography based on Hilbert space and Von Neumann quantum mechanics, excluding the effects of decoherence. We show that under specific conditions, the rigged Hilbert space formulation for quantum cryptography reduces to the Hilbert space formulation. However, there are opportunities for an eavesdropper on a secure channel to avoid detection using extended space techniques in the generalized functional space. In the generalized functional space, non-unitary operators can be constructed, that act on non-orthogonal mixed states without isometry; in this case, the no-cloning theorem for two non-orthogonal states is less efficient. Thus, functional space may provide a means for successfully implementing quantum cryptography.