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.
The most general duality gates were introduced by Long,Liu and Wang and named allowable generalized quantum gates (AGQGs,for short).By definition,an allowable generalized quantum gate has the form of U=YfkjsckUK,where Uk’s are unitary operators on a Hilbert space H and the coefficients ck’s are complex numbers with |Yfijo ck\ ∧ 1 an d 1ck| <1 for all k=0,1,...,d-1.In this paper,we prove that an AGQG U=YfkZo ck∧k is realizable,i.e.there are two d by d unitary matrices W and V such that ck=W0kVk0 (0<k<d-1) if and only if YfkJt 1c*|<m that case,the matrices W and V are constructed.
