The Study Of Entanglement Measures Of Channel,Orthogonal Product Set With Strong Nonlocality And Optimal Convex Approximation Of Qubit States

Quantum entanglement,as a fundamental property of quantum mechanics,is widely used in quantum teleportation,quantum key distribution,quantum state discrimination and entanglement-assisted communication and so on.Entanglement of quantum channel is a kind of generalized entanglement resource,which refers to its ability to generate or increase entanglement of quantum states.Understanding and utilizing various forms of quantum resources is a subject of quantum information science,while quantifying the entanglement resources of channel is an important branch of this subject,which can provide a powerful theoretical evidence for the realization of quantum technology.Recently,great progress has been made on the study of strong quantum nonlocality without entanglement.A large number of orthogonal product sets with strong nonlocality have been proposed.Their local indistinguishability has important applications in quantum data hiding and quantum secret sharing.However,some open problems remain unsolved,such as,the issue of constructing strongly nonlocal orthogonal product set with fewer elements and the problem of the existence of orthogonal product sets with strong quantum nonlocality in any even party systems.As for strongly nonlocal orthogonal product set,another interesting phenomenon is that its indistinguishability can be changed by sharing additional entanglement resources among the parties.Since entanglement is an expensive resource,it is very significant about how to use less entanglement resources to accomplish the local discrimination tasks of these sets.With respect to the quantum states in reality,it is also a subject worthy of study to optimally approximate a target state by the convex combination of a set of available states,because there are always some states that are not easily obtained directly.In addition,it is worth noting that geometry has some characteristics,including intuitive and easy to understand.It is the characteristic of this dissertation to associate the research content with geometry.In this paper,we mainly study three aspects:entanglement measure of channel,orthogonal product set with strong quantum nonlocality and optimal convex approximation of qubit state.The whole dissertation consists of five chapters.In chapter 1,we review some necessary basic symbols and concepts,including quantum states,quantum channels,density operators,measurement operators,etc.In chapter 2,we first briefly introduce the entanglement resource theory of channel.whereafter,for a quantum system of any finite dimension,we define a distance measure between channels,Choi relative entropy of channels,and propose three new entanglement measures of channels under the framework of resource theory,according to the Choi relative entropy of channels,concurrence and k-ME concurrence,respectively.Rigorous proofs show that our newly presented entanglement measures fulfill all the requirements dictated by the resource theory of quantum channels,that is,these entanglement measures satisfy non-negativity,weak monotonicity,strong monotonicity,convexity,additivity and subadditivity.Meanwhile,we describe the relationship between the hypothesis testing and the entanglement measure of channels,and illustrate an obvious advantage of these measures in computability compared with other known entanglement measures of quantum channels,and classify quantum channels based on the power of generating entanglement.In addition,we also give several detailed examples to characterize the properties of quantum channels and the conversions between quantum coherence states and entangled states and between fully separable states and multipartite entangled states.Chapters 3 and 4 discuss the orthogonal product set with strong quantum nonlocality in multipartite quantum systems.In chapter 3,based on the plane stucture of orthogonal product set,we propose a sufficient condition which is enough to show that the orthogonality-preserving positive operator-valued measures performed on fixed subsystem can only be trivial and partially answer an open question given by Yuan et al.in Ref.[Phys.Rev.A 102,042228(2020)],"Can we find the smallest strongly nonlocal set in C3(?)C3(?)C3,and more generally in any tripartite systems?".Moveover,by using the connection between the nonlocality and the plane structure of orthogonal product set,we construct a strongly nonlocal orthogonal product set in CdA(?)CdB(?)CdC(dA,dB,dC≥ 4),which contains fewer quantum states,and generalize the structures of known orthogonal product sets to any possible three and four-partite systems.In addition,we present several entanglement-assisted protocols for perfectly local discrimination of the sets.It is shown that the protocols without teleportation use less entanglement resources on average and these sets can always be discriminated locally with multiple copies of two-qubit maximally entangled states.These results also exhibit nontrivial signification of maximally entangled states in the local discrimination of quantum states.In chapter 4,we successfully construct strongly nonlocal orthogonal product sets in n-partite systems for all even n,which answers the open questions given by Halder et al.[Phys.Rev.Lett 122,040403(2019)]and Yuan et al.[Phys.Rev.A 102,042228(2020)]for any possible even party systems.Thus,we find general construction of strongly nonlocal orthogonal product sets in space(?)i=1nCdi(n,di≥ 3)and show that there do exist incomplete orthogonal product bases that can be strongly nonlocal in any possible n-partite systems for all even n.We analyze the differences and connections between these sets and the known orthogonal product sets in odd party systems.In addition,we present a local state discrimination protocol for our sets by using additional entangled resource.When at least two subsystems have dimensions greater than three,the protocol consumes less entanglement than teleportation-based protocol.Strongly nonlocal set implies that the information cannot be completely accessed as long as it does not happen that all parties are together.As an application,we connect our sets with local information hiding in multipartite system.In chapter 5,we investigate the optimal convex approximation,optimally approximating a desired and unavailable qubit state by the convex mixing of a given set of available states.When the available states are the eigenvectors of three Pauli matrices,we present the complete exact solution for the optimal convex approximation of an arbitrary qubit state based on the fidelity distance.By the comparison of optimal states based on fidelity and trace norm,the advantages and disadvantages of the optimal convex approximation are identified.Several specific examples are provided to support this.We also analyze the geometrical properties of the target states which can be completely represented by a set of available states.Such a target state is also called CR(completely represented)state.Using the feature of convex combination,we derive the maximum range of CR states and clearly illustrate that any qubit state can be optimally prepared by at most three available states in known available states.