An allowable generalized quantum gate (introduced by Long, Liu and Wang) has the form of 1 0 d k k k U cU − = = , ∑ where U k 's are unitary operators on a Hilbert space H and 1 0 1 d k k c − = | | ≤ ∑ and 1 k c | |≤ (0≤k≤d−1). In this work we consider a kind of AGQGs, called restricted allowable generalized quantum gates (RAGQGs), satisfying 1 0 0 1 d k k c − = < | |≤ . ∑ Some properties of the set RAGQG(H) of all RAGQGs on H are established. Especially, we prove that the extreme points of RAGQG(H) are exactly unitary operators on H and that B(H)=R + RAGQG(H). duality computer, Hilbert space, unitary, quantum gate, restricted allowable generalized quantum gate Citation: Cao H X, Li L, Chen Z L, et al. Restricted allowable generalized quantum gates.

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Denoted by M(A), QM(A) and SQM(A) the sets of all measures, quantum measures and subadditive quantum measures on a σ-algebra A, respectively. We observe that these sets are all positive cones in the real vector space F(A) of all real-valued functions on A and prove that M(A) is a face of SQM(A). It is proved that the product of m grade-1 measures is a grade-m measure. By combining a matrix M μ to a quantum measure μ on the power set A n of an n-element set X, it is proved that μ ν ≺≺ (resp. μ ν ⊥ ) if and only if μ ν ≺≺ M M (resp. M μ M v =0). Also, it is shown that two nontrivial measures μ and ν are mutually absolutely continuous if and only if μ·ν∈QM(A n ). Moreover, the matrices corresponding to quantum measures are characterized. Finally, convergence of a sequence of quantum measures on A n is introduced and discussed; especially, the Vitali-Hahn-Saks theorem for quantum measures is proved. quantum measure, absolute continuity, product, matrix representation, convergence Citation: Guo Z H, Cao H X, Chen Z L, et al. Operational properties and matrix representations of quantum measures.

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