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...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.展开更多
Letϕ:Pc(M1)→Pc(M2)be a surjective Lp-isometry between Grassmann spaces of projections with the trace value c in semifinite factors M1 and M2.Based on the characterization of surjective Lp-isometries of unitary groups...Letϕ:Pc(M1)→Pc(M2)be a surjective Lp-isometry between Grassmann spaces of projections with the trace value c in semifinite factors M1 and M2.Based on the characterization of surjective Lp-isometries of unitary groups in finite factors,we show thatϕor I−ϕcan be extended to a∗-isomorphism or a∗-antiisomorphism.In particular,ϕis given by a∗-(anti-)isomorphism unless M1 and M2 are finite and c=12.展开更多
A norm of a quaternion matrix is defined. The expressions of the least square solutions of the quaternion matrix equation AX = B and the equation with the constraint condition DX = E are given.
In this paper, we show that a multiplication operator on the Dirichlet space D is unitarily equivalent to Dirichlet shift of multiplicity n ＋ 1 （n ≥ 0） if and only if its symbol is c zn＋1 for some constant c. The...In this paper, we show that a multiplication operator on the Dirichlet space D is unitarily equivalent to Dirichlet shift of multiplicity n ＋ 1 （n ≥ 0） if and only if its symbol is c zn＋1 for some constant c. The result is very different from the cases of both the Bergman space and the Hardy space.展开更多
Let F<sub>q<sup>2</sup></sub> be a finite field with q<sup>2</sup> elements, where q is a power of a prime and let a→a=a<sup>q</sup> （1） be the involutive automorphis...Let F<sub>q<sup>2</sup></sub> be a finite field with q<sup>2</sup> elements, where q is a power of a prime and let a→a=a<sup>q</sup> （1） be the involutive automorphism of F<sub>q<sup>2</sup></sub> with the fixed field F<sub>q</sub>. An n×n matrix H over F<sub>q<sup>2</sup></sub> is said to be Hermitian, if H<sup>T</sup>=H, where H is the matrix obtained from H by replacing every dement of H by its image展开更多
Based on a new characterization of bounded and compact weighted compositionoperators on the Fock space obtained by Le T （Le T. Normal and isometricweighted composition operators on the Fock space. Bull. London. Math....Based on a new characterization of bounded and compact weighted compositionoperators on the Fock space obtained by Le T （Le T. Normal and isometricweighted composition operators on the Fock space. Bull. London. Math. Soc., 2014,46： 847-856）, this paper shows that a bounded weighted composition operator onthe Fock space is a Fredholm operator if and only if it is an invertible operator, andif and only if it is a nonzero constant multiple of a unitary operator. The result isvery different from the corresponding results on the Hardy space and the Bergmanspace.展开更多
The goal of this paper is to confirm that the unitary group U(H) on an infinite dimensional complex Hilbert space is a topological group in its strong topology, and to emphasize the importance of this property for app...The goal of this paper is to confirm that the unitary group U(H) on an infinite dimensional complex Hilbert space is a topological group in its strong topology, and to emphasize the importance of this property for applications in topology. In addition, it is shown that U(H) in its strong topology is metrizable and contractible if H is separable. As an application Hilbert bundles are classified by homotopy.展开更多
基金supported by the National Natural Science Foundation of China (Grant Nos. 10571113 and 10871224)the Natural Science Research Program of Shaanxi Province (Grant No. 2009JM1011)
文摘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.
基金supported by the Science and Technology Research Program of Chongqing Municipal Education Commission(Grant No.KJQN2021000529)the Natural Science Foundation of Chongqing Science and Technology Commission(Grant No.cstc2020jcyj-msxm X0723)+2 种基金supported by Young Talent Fund of University Association for Science and Technology in Shaanxi(Grant No.20210507)supported by National Natural Science Foundation of China(Grant Nos.11871127and 11971463)supported by National Natural Science Foundation of China(Grant Nos.11971463,11871303 and 11871127)。
文摘Letϕ:Pc(M1)→Pc(M2)be a surjective Lp-isometry between Grassmann spaces of projections with the trace value c in semifinite factors M1 and M2.Based on the characterization of surjective Lp-isometries of unitary groups in finite factors,we show thatϕor I−ϕcan be extended to a∗-isomorphism or a∗-antiisomorphism.In particular,ϕis given by a∗-(anti-)isomorphism unless M1 and M2 are finite and c=12.
文摘A norm of a quaternion matrix is defined. The expressions of the least square solutions of the quaternion matrix equation AX = B and the equation with the constraint condition DX = E are given.
基金Supported by Tianyuan Foundation of China (Grant No. 10926143)Young Science Foundation of Shanxi Province(Grant No. 2010021002-2)+2 种基金the National Natural Science Foundation of China (Grant No. 10971195)the Natural Science Foundation of Zhejiang Province (Grant Nos. Y6090689 Y6110260)
文摘In this paper, we show that a multiplication operator on the Dirichlet space D is unitarily equivalent to Dirichlet shift of multiplicity n ＋ 1 （n ≥ 0） if and only if its symbol is c zn＋1 for some constant c. The result is very different from the cases of both the Bergman space and the Hardy space.
文摘Let F<sub>q<sup>2</sup></sub> be a finite field with q<sup>2</sup> elements, where q is a power of a prime and let a→a=a<sup>q</sup> （1） be the involutive automorphism of F<sub>q<sup>2</sup></sub> with the fixed field F<sub>q</sub>. An n×n matrix H over F<sub>q<sup>2</sup></sub> is said to be Hermitian, if H<sup>T</sup>=H, where H is the matrix obtained from H by replacing every dement of H by its image
文摘Based on a new characterization of bounded and compact weighted compositionoperators on the Fock space obtained by Le T （Le T. Normal and isometricweighted composition operators on the Fock space. Bull. London. Math. Soc., 2014,46： 847-856）, this paper shows that a bounded weighted composition operator onthe Fock space is a Fredholm operator if and only if it is an invertible operator, andif and only if it is a nonzero constant multiple of a unitary operator. The result isvery different from the corresponding results on the Hardy space and the Bergmanspace.
文摘The goal of this paper is to confirm that the unitary group U(H) on an infinite dimensional complex Hilbert space is a topological group in its strong topology, and to emphasize the importance of this property for applications in topology. In addition, it is shown that U(H) in its strong topology is metrizable and contractible if H is separable. As an application Hilbert bundles are classified by homotopy.
