Recently, Gudder proved that the set of all generalized quantum gates coincides the set of all contractions in a finite-dimensional Hilbert space (S. Gudder, Int. J. Theor. Phys. 47:268-279, 2008). In this note, we proved that the set of all generalized quantum gates is a proper subset of the set of all contractions on an infinite dimensional separable Hilbert space H. Meanwhile, we proved that the quantum operation deduced by an isometry is an extreme point of the set of all quantum operations on H.Keywords Generalized quantum gate · Duality quantum computer · Extreme point of quantum operations

In this paper, releasing the restriction on operators which are self-adjoint, we propose a Heisenberg-type uncertainty relation and a Schrödinger-type uncertainty relation with any pair of operators on a Hilbert space. A generalization of Luo's theorem [S. Luo, “Heisenberg uncertainty relation for mixed states,” Phys. Rev. A 72, 042110 (2005)] is investigated.

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In this paper, the closedness of ranges and Kato non-singularity of upper triangular operator matrices are investigated.Crown