Mathematical theory of the duality computer in the density matrix formalism

Very recently, Gudder has given the mathematical theory for the duality computer [1]. In his timely work, he has provided two descriptions, one is in the pure state formalism and the other is in the mixed state formalism for the mathematics of duality computer. The duality computer is a new type of computing machine based on the general principle of quantum interference proposed in Ref.[2]. Notably, unsorted database search problem may be solved by using only a single query, and all NPcomplete problems may have polynomial algorithms in the duality computer[2]. In Ref.[1], it has been pointed out that a paradoxical situation occurs with the mixed state situation where the formalism gives a different result from the pure state formalism, and in particular, the advantage of duality computer was lost in the mixed state formalism. In this paper, we give the mathematical theory of the duality computer in the density matrix formalism. This density matrix formalism description is in accordance with Gudder’s pure state description, and hence solves the paradox. Density matrix may have different physical interpretations [3, 4], and different interpretation may lead to different results if issues concerning the fundamentals are involved. In this short paper, we can either consider the density matrix arises from the average of many computations using duality computers starting from different initial states, or consider it as an improper mixed state arising from the average over other unseen degrees of freedom. Here we just study the mathematical formalism and do not consider fundamental issues where difference may arise.