Quantum Cloning Based On The Maximin Principle

No-cloning theorem is one of the basic results in quantum mechanics and quantum information,an unknown quantum state can not be cloned perfectly.However no-cloning theorem is a qualitative theory,and for the further application,quantitative investigation are necessary.Nowadays,the research of quantum cloning method is mainly divided into two categories:one is the approximate quantum cloning,and the other is the probabilistic quantum cloning.In recent years,much progress has been made in studying quantum cloning,both theoretically and experimentally.But these researches are focused on the cloned input set which is special,it is quite necessary to exploit quantum cloning theory on the more general input set.This thesis studies approximation quantum cloning on the more general distribution of pure state and mixed state,by using the Maximin principle.The main contributions are presented as follows:First,the pure state quantum cloning has been studied.We propose a novel optimal1→2 quantum cloning method based on the Maximin Principle by making full use of a priori information of amplitude and phase about the general cloned qubit input set.To in-vestigate more general cloning machine inputs,we describe a priori information of these input sets with“longitude-latitude boundary”on the Bloch sphere.By using our new method,we explore both the optimal cloning on sphere hat input set and the sub-optimal quantum cloning on arbitrary general set closed by“longitude-latitude grid”on the Bloch sphere,and their analytic solutions are obtained respectively.Through comparison and analysis,it is shown that the fidelity of our optimal sphere hat cloning machine is the largest one among all the cloning machines.In addition,it is revealed that two most fa-mous quantum cloning machines,universal cloning machine and phase-covariant cloning machine,can be considered as special cases of our sub-optimal cloning machine.More-over,our sub-optimal cloning machine also outperforms two aforementioned cloning ma-chines since there is more a priori information about the input set available.Additionally,the quality of our sub-optimal quantum cloning machine is better than the belt quantum cloning evaluated with a fidelity in many cases.Second,the mixed state quantum cloning is further explored.Since there are noises in reality,the exploration on the mixed state quantum cloning machine is more practical.It takes three parameters to describe a mixed state set:amplitude,phase and radius.We investigate the optimal quantum cloning for a mixed state set in the equatorial plane of the Bloch sphere,and obtain an explicit formula for it.This suggests that our proposed new method can also be applied to the mixed state cloning.Moreover,when the radius of input set is 1,our optimal equatorial plane quantum cloning is reduced to phase-covariant quan-tum cloning.In addition,we put forward a new decomposition of the high-dimensional identical mixed quditsρ^（?）N,and present a d-dimensional N→M mixed state universal quantum broadcasting whose shrink factor reaches the optimal bound.We extend the previous works from 2-dimensional N→M mixed state universal quantum broadcasting to d-dimensional one.Finally,some further discussions are given in the last chapter.