Approximation Of Quantum Assemblages And α-Fidelity Of Some Classes Of Quantum States

In quantum information theory,people usually consider the following two important questions:(1)What does it mean that two information are similar?(2)What kind of process can transmit information? The approximation of quantum assemblages and the α-fidelity discussed in this thesis are closely related to these questions and provide qualitative and quantitative answers to them.This thesis consider the realization and approximation of quantum assemblages in quantum steering scenario.On the other hand,to analyze the degree of change of the quantum state after the evolution of the quantum channels and to quantify the similarity of Gaussian states by α-fidelity.The main work of this thesis is as following:Firstly,the quantum realization and approximation of quantum assemblages are studied.On the one hand,this thesis demonstrated that there is a quantum assemblage without quantum realization while the local dimension is infinite and the input and output are not equal to 2.That is,there exists an assemblage in infinite dimension which doesn’t have a quantum realization.Indeed,it is a reformulation of Tsirelson’s problem in the context of the quantum steering.On the other hand,this thesis provided a method to approximate the assemblages in finite-dimensional Hilbert spaces and use it to construct a set of projectors on a 2n dimensional Hilbert space which can universally work for all quantum assemblages with binary inputs and outputs.The dimension 2n of the Hilbert space depends on the accuracy of the approximation.Secondly,the α-fidelity extremum problem for unitary orbits is analyzed.The analytical formulas for maximum and minimum values of quantum α-fidelity under unitary orbits of quantum states are explicitly derived by applying rearrangement inequalities,matrix trace inequalities,and theory of majorization,and show that for two quantum states ρand σ with some relations,the extremum of α-fidelity can be obtained.Furthermore,theα-fidelity is successfully verified to go through the whole closed interval,which works from the minimum value to the maximum value.As a corollary,the sandwiched α-Rényi relative entropy of two unitary orbits can be described by its classical form.Thirdly,this thesis considered quantum fidelity of two states through a quantum channel and give a geometric interpretation.This thesis obtained the maximum and minimum of α-fidelity which takes all arbitrary channels,all unital channels,and all mixed unitary channels.Further,this thesis given a geometric interpretation of the α-fidelity about the closed and convex sets of density matrices and quantum channels.That is,α-fidelity can be taken as a geometric measure of distance between two spaces.Finally,the α-fidelity of the multi-dimensional Gaussian states with diagonalizable second-order Hamiltonian is investigated.The quantum α-fidelity of two multi-mode Gaussian states is accurately calculated by using a matrix representation method.To verify an universality of the conclusion,three typical examples of decomposable and nondecomposable systems,including one-modedisplaced-squeezed thermal states,two-mode displaced-squeezed thermal states in the decomposable system and Liu-Tombesi oscillators in the non-decomposable system,respectively.