The Study On Quantum Measurements,entropy Divergences And Operator Convex Functions

In this thesis,we study quantum information from three aspects: joint measurability of quantum measurements,symmetric quantum Jensen divergences and operator strictly convex functions.The thesis is separated into the following four chapters:The first chapter is the introduction,which describes symbol explanations and preliminaries,etc.Firstly,the research background and current situation of joint quantum measurements,symmetric quantum Jensen divergences and operator strictly convex functions are introduced;secondly,some symbols used in this thesis are explained;finally,some basic concepts and well-known facts involved in this research content are given,including the definitions and properties of quantum joint measurements,quantum entropy divergences,symmetric quantum Jensen divergences and operator strictly convex functions.The second chapter is the joint measurability of quantum measurements,which mainly studies the joint measurability of a pair of positive operator valued measures.It is known that a pair of quantum measurements A and B are jointly measurable.By replacing the elements in A and B,a new pair of quantum measurements C and D are constructed.By using the definition of joint measurability,a joint measurement of quantum measurements C and D is found,and it is proved that C and D are jointly measurable.In addition to element replacement,two new sets of quantum measurements C and D are constructed by recombining the elements in quantum measurements A and B.By using similar proof methods,it is proved that the two new sets of quantum measurements are jointly measurable.The third chapter is the symmetric quantum Jensen divergences,which mainly studies the lower bounds of three kinds of symmetric quantum Jensen divergences.The quantum Jensen-Shannon divergence,quantum Jensen-Shannon-Tsallis divergence and quantum Jensen-Shannon-Reny(?) divergence are corresponding to von Neumann,Tsallis and Reny(?) entropies respectively.By using an inequality for the convexity of a kind of operator functions and the integral representation of power functions,the lower bounds of the three kinds of quantum Jensen divergences are given via evaluating a second derivative of some function.The fourth chapter is the operator strictly convex functions,which mainly studies the strict convexity of two kinds of operator convex functions.Strictly convex functions are closely related to quantum entanglement,and they play an important role in determining whether entanglement measure satisfies monogamy.For two kinds of operator convex functions involved in commutative complex positive definite density matrix set and complex positive definite density matrix set,an inequality about the convexity of a kind of operator functions and the integral representation of power function are used to study,and the conditions of these two kinds of operator convex functions satisfying strict convexity are given respectively.