Using the technique of block-operators, in this note, we prove that if P and Q are idempotents and （P - Q）^2n＋1 is in the trace class, then （P - Q）^2m＋1 is also in the trace class and tr（P - Q）^2m＋1 = dim（k（P） ∩ k（Q）^⊥） -dim（k（P）^⊥ N k（Q））, for all m ≥ n. Moreover, we prove that dim（k（P）∩ k（Q）^⊥） = dim（k（P）^⊥ ∩k（Q）） if and only if there exists a unitary U such that UP = QU and PU = UQ, where k（T） denotes the range of T. Keywords Fredholm, orthogonal projection, positive operator
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.
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 Mμto a quantum measureμon the power set An of an n-element set X,it is proved thatμν(resp. μ⊥ν)if and only if μν M M(resp.MμMv=0).Also,it is shown that two nontrivial measuresμandνare mutually absolutely continuous if and only ifμ·ν∈QM(An).Moreover,the matrices corresponding to quantum measures are characterized. Finally,convergence of a sequence of quantum measures on An is introduced and discussed;especially,the Vitali-Hahn-Saks theorem for quantum measures is proved.
