We derive that sufficient and necessary conditions for existence of a quantum channel φ and a generalized unitary operation ε sending Ai to Bi(1≤ i≤ k) for two given families {Ai}i=1k,{Bi}i=1k of matrices,respectively.As an application,a sufficient and necessary condition for existence of a unitary duality quantum computer with given input-output states is obtained.
In this paper, we derive an upper bound for the adiabatic approximation error, which is the distance between the exact solution to a Schrodinger equation and the adiabatic approximation solution. As an application, we obtain an upper bound for 1 minus the fidelity of the exact solution and the adiabatic approximation solution to a SchrOdinger equation.
Two linear In this letter, we prove the following conclusions by introducing a function Fn（t）： （1） If a quantum system S with a time-dependent non-degenerate Hamiltonian H（t） is initially in the n-th eigenstate of H（0）, then the state of the system at time t will remain in the n-th eigenstate of H（t） up to a multiplicative phase factor if and only if the values Fn（t） for all t are always on the circle centered at 1 with radius 1; （2） If a quantum system S with a time-dependent Hamiltonian H（t） is initially in the n-th eigenstate of H（0）, then the state of the system at time t will remain c-uniformly approximately in the n-th eigenstate of H（t） up to a multiplicative phase factor if and only if the values F,（t） for all t are always outside of the circle centered at 1 with radius 1-ε. Moreover, some quantitative sufficient conditions for the state of the system at time t to remain ε-uniformly approximately in the n-th eigenstate of H（t） up to a multiplicative phase factor are established. Lastly, our results are illustrated by a spin-half particle in a rotating magnetic field.
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.
Let B（Н） be the algebra of all the bounded linear operators on a Hilbert space Н. For A, P and Q in B（Н）, if there exists an operator X ∈ B（Н） such that APXQA = A, XQAPX = X, （QAPX）^＊ = QAPX and （XQAP）^＊ = XQAP, then X is said to be the F-inverse of A associated with P and Q, and denoted by A^＋P,Q. In this note, we present some necessary and sufficient conditions for which A^＋P,Q exists, and give an explicit representation of A^＋PQ （if A^＋P,Q exists）.
