Quantum Hypothesis Testing Relative Entropy And Applications

Quantum information processing is a research field that combines quantum mechanics,mathematics,information theory and other disciplines,and enriches the research content of information theory.As the basis of quantum information processing,discrimination of quantum states and quantum hypothesis testing are closely related.In quantum hypothesis testing,there are two different methods to deal with the two types of errors probabilities arising from the testing,that is symmetrical and asymmetrical hypothesis testing.In symmetric hypothesis testing,one treats the errors of first and second kind in a symmetric way,which is the weighted average sum of the two kinds of error probabilities,while asymmetric hypothesis testing that limits the one type of error probability and minimizes the other type of error probability,as a result that the related research has also been well applied in other fields.In general,the states involved in quantum hypothesis testing are in the form of asymptotically multiple copies,but in actual processing,independent and identically distributed quantum states are difficult to prepare and operations on larger systems are also difficult to achieve,thus the quantum hypothesis testing in the non-asymptotic case is taken into consideration.This thesis has discussed quantum asymmetry hypothesis testing in non asymptotic case,and based on the idea of limiting the first type of error probability and minimizing the second type of error probability emits the definition of quantum hypothesis testing relative entropy.This thesis is divided into two parts,the first part provides a comprehensive and systematic study on the definitions and properties of the quantum hypothesis testing relative entropy,including discussing the relations between the quantum hypothesis testing relative entropy and other entropy measures,as well as the applications in quantum coherent resource theory.In the random numbers,the quantum hypothesis testing conditional entropy and its properties have been good results,but there are relatively few discussions about the form of von Neumann mutual information.According to the properties of quantum hypothesis testing relative entropy and conditional entropy,the second part of this thesis has discussed the definition of quantum hypothesis testing mutual information and given the related properties.It is found that there is no chain relation,which holds for von Neumann entropy,on this type of mutual information,consequently by constructing a measure,the chain relation of quantum hypothesis testing mutual information is discussed.