Complex duality quantum computers acting on pure and mixed states

The aim of this paper is to establish a mathematical fundamental of complex duality quantum computers (CDQCs) acting on vector-states (pure states) and operator-states (mixed states), respectively. A CDQC consists of a complex divider, a group of quantum gates represented by unitary operators, or quantum operations represented by completely positive and trace-preserving mappings, and a complex combiner. It is proved that the divider and the combiner of a CDQC are an isometry and a contraction, respectively. It is proved that the divider and the combiner of a CDQC acting on vector-states are dual, and in the finite dimensional case, it is proved that the divider and the combiner of a CDQC acting on operator-states (matrix-states) are also dual. Lastly, the loss of an input state passing through a CDQC is measured.