The Study Of Quantum Uncertainty Relations And Multilevel Coherence

Quantum uncertainty relation and quantum coherence are two of the most fundamental characteristics of quantum mechanics discriminating from classical physics.Both of them have important applications in quantum information science,quantum metrology,etc.Quantum uncertainty relation and quantum coherence have been studied for a long time,and are still important subjects in the field of physics.Quantum uncertainty relation is one of the most basic laws obeyed by the quantum world.It represents and measures the uncertainty of quantum measurement.Uncertainty relations have many practical applications in the field of quantum information,such as quantum entanglement witness,quantum random number generation,quantum key distribution,quantum cryptography,etc.There exist many forms of uncertainty relations,such as standard-deviation-based uncertainty relations,entropy uncertainty relations,skew-information-based uncertainty relations,coherence-based uncertainty relations,etc.It is difficult to optimize the lower bound of any form of uncertainty relation,but it is an important subject in studying uncertainty relation.On the other hand,quantum coherence originates from the superposition principle of quantum states and is the basis of quantum interference,nonlocality and quantum entanglement.Quantum coherence is now recognized as a relatively fledged physical resource,and plays a central role in fundamental physics.It is widely used in various information processing tasks,such as quantum cryptography,metrology,thermodynamics,and quantum biology.Despite a great deal of recent progress,however,the majority of current literature focuses on a rather coarse grained description of coherence,which is ultimately insufficient to reach a complete understanding of the fundamental role of quantum superposition in the afore-mentioned tasks.To overcome such limitations,one needs to take into consideration the number of classical states in coherent superposition-contrasted with the simpler question of whether any nontrivial superposition exists-which gives rise to the concept of multilevel quantum coherence.Deciphering this structure can yield a tangible impact on many areas of physics,such as statistical mechanics,condensed matter,and transformation phenomena in many-body systems.So the study of multilevel quantum coherence has important physical significance.In this paper,we focus on the study of quantum uncertainty relations and multilevel quantum coherence.We obtain some new uncertainty relations based on Wigner-Yanase skew information for finite observables and channels,and proof that our uncertainty relations have tighter lower bounds than the existing ones.In the resource theory of multilevel quantum coherence,we mainly study the transformations of resource pure states via free operations,and we present the condition of the interconversions of two multilevel coherent resource pure states under k-coherence-preserving operations.In the end,we focus on the issue of entanglement,then obtain the relationship between k-ME concurrence and negativity for multipartite quantum systems.The content of the whole dissertation is divided into the following four parts:In Chapter 1,we briefly review some basic concepts related to the research con-tent,including mathematical symbols,(reduced)density operators,quantum operations,quantum channels and so on.The main results of this paper are contained in Chapters 2,3 and 4.In Chapter 2,we study the uncertain relations based on Wigner-Yanase skew information for finite observables and channels.Firstly,we review the standard-deviation-based uncertainty relations and entropy uncertainty relations.Then,the main results of this chapter are given.By using the norm inequality of vector space,we establish a new uncertainty relation in terms of skew information for n Hermitian operators,which is saturated(thus it holds as equality)for two incompatible observables.Our uncertainty relations have tighter lower bounds than the existing ones.Detailed examples are provided.At last,we present two uncertainty relations for arbitrary finite quantum channels by using skew information.We hope that our method can be widely applied to the study of uncertainty relations for observables and channels.In Chapter 3,we introduce the frameworks of coherence and multilevel coherence,at the beginning.Then,we study the transformation of multilevel coherent resource pure states via the free operations.In the resource theory of multilevel quantum coherence,suppose free operations are k-coherence-preserving operations(k? {1,2,…,d}).We prove that,any two resource pure states can be interconverted with a nonzero probability under a k-coherence-preserving map.The probability is related to the robustness and geometric measure of multilevel quantum coherence,and can be estimated effectively.This is useful as this bound is often not easy to compute for other resources,e.g.in multipartite entanglement theory.Meanwhile,we present the condition of the interconversions of any two multilevel coherent resource pure states under k-coherence-preserving operations.In addition,we proof that in the resource theory of multilevel quantum coherence,no resource state is isolated.That is,given a resource pure state |?>,there always exists another resource pure state |?>,and a k-coherence-preserving operation such that |?>can be transformed to |?>via the k-coherence-preserving operation.Finally,we obtain that the coherence state(?)can be transformed to any other multilevel coherent resource state via k-coherence-preserving operations.In Chapter 4,we investigate the issue of quantum entanglement.After a brief review of quantum entanglement,we study the relation between k-ME concurrence and negativity.For all the n-qubit pure states,we obtain an equal relation between n-ME concurrence and negativity.While for any n-qubit mixed state,there is an inequation between n-ME concurrence and negativity.At last,we give an exact value of the n-ME concurrence for a mixed state,using the relation between n-ME concurrence and negativity.