Study On Quantum Coherence And The Quantumness Of Quantum Ensembles

Quantum information theory is an interdisciplinary discipline generated from the combination of quantum theory and information science,which has broad prospects in both theoretical research and technical applications.The quantification and characterization of quantum coherence is an frontier topic in the field of quantum information science.Quantum coherence is an important physical resource,which plays an indispensable role in many quantum information processing tasks,and has wide applications in quantum metrology,quantum biology and quantum optics.However,an ideal closed quantum system does not exist,a real quantum system inevitably interacts with its surrounding environment,which leads to decoherence phenomena.Quantum uncertainty relation is one of the most basic principles of quantum mechanics.Quantum uncertainty relations have many applications in quantum information theory,such as quantum randomness,quantum key distribution and quantum cryptography.Quantum coherence can also be exploited to investigate the quantumness of quantum ensembles.The study on quantum coherence and quantum uncertainty relations can not only help us deeper the understanding of quantum features,but also distinguish the boundaries between quantum and classical world.In this thesis,we mainly study the evolution of quantum coherence,quantum uncertainty relations and the quantumness of quantum ensembles.The thesis is divided into four chapters.In Chapter 1,we briefly introduce the background and status of the topics of this thesis and some preliminaries of quantum information theory.In Chapter 2,we study the evolution properties of coherence for quantum states under quantum channels via n-th decay rate.We first define the concept of n-th decay rate of quantum coherence,and then derive the analytical formulas of the n-th decay rate for Bell diagonal states under several kinds of quantum channels based on different coherence measures.Finally,we investigate the geometry of Bell-diagonal states when the coherence is frozen.It is shown that when n is large enough,the coherence of Belldiagonal states will not be frozen under generalized amplitude damping channels,depolarization channels and phase flip channels.In Chapter 3,we discuss uncertainty relations and complementarity relations based on generalized Wigner-Yanase-Dyson skew information.We first propose the concept of quantum uncertainty utilizing generalized Wigner-Yanase-Dyson skew information.Based on it,we then derive several uncertainty relations and complementarity relations with respect to a set of mutually unbiased measurements and general symmetric informationally complete positive operator valued measurements.Our results include some existing ones as particular cases.In Chapter 4,we use the coherence of a quantum ensemble to investigate the quantumness of a quantum ensemble.First,we give the definition of generalized?-z relative Rényi entropy between ensembles.Then we define a quantifier of the coherence of ensembles via generalized?-z relative Rényi entropy,and use it to further quantify the quantumness of a quantum ensemble,which generalize the measure of quantumness defined via relative entropy.It is proved that this measure satisfies the intuitive and desirable properties,such as unitary invariance,positivity,monotonicity under any commutativity preserving operations and so on.