Mathematics Colloquium, Fall 2007

Impurity and quaternions in nonrelativistic scattering from quantum memory

Models in quantum computing rely on transformations of states of quantum memory. We study mathematical aspects of a model proposed by Wu in which the memory state is changed via scattering of incoming particles. This operation causes the memory content to deviate from a pure state, i.e. induces impurity. For nonrelativistic particles scattered from a two-state memory and sufficiently general interaction potentials in 1 + 1 dimensions, we express impurity in terms of quaternionic commutators. In this context, pure memory states correspond to null hyperbolic quaternions. In the case of point interactions, the scattering process amounts to appropriate rotations of quaternions in the frequency domain. This point of view complements a previous analysis by Margetis and Myers (2006 J. Phys. A 39: 11567-11581) and is in collaboration with D. Margetis.