Quantum Correlations In Tree-shaped Quantum Networks

Operator theory and operator algebra have profound backgrounds in quantum mechanics and have direct applications in quantum information sciences,which results in more and more scholars in the field of operator theories and operator algebras devoting themselves to the related research fields of quantum information sciences.The existences of quantum corre-lations in quantum networks are fundamental for long-distance quantum communications.In recent years,the study of quantum correlations in quantum networks has become one of the hot topics in the cross field-s of operator theory and quantum information science.In this thesis,by using theories and methods of operator theory and operator algebra,we examine quantum correlations,such as nonlocality,quantum entan-glement and EPR steering,in the tree-shaped quantum networks with different structures.The following are the main results of this thesis.1.The nonlocality of the two-forked tree-shaped quantum network with depth two is investigated,where the network includes（2ⁿ-1）（n≥3）parties and（2ⁿ-2）independent sources,and each party has two inputs and two outputs.A（2ⁿ-2）-local inequality that this network necessar-ily satisfies is obtained.It is shown that if each source in the network produces two-qubit entangled pure states,they necessarily violate this in-equality.If each source in the network produces two-qubit mixed states,a necessary condition for them violating this inequality is derived.Based on these results,the nonlocality of any-forked tree-shaped quantum net-works with depth two and each party having two inputs and two outputs is obtained.2.The nonlocality of the two-forked tree-shaped quantum network with depth two and containing seven parties and six independent sources is discussed.One case is each party having arbitrary m（m≥2）inputs and two outputs,while the other case is every extreme party having ar-bitrary m（m≥2）inputs with two outputs and every intermediate party having 2^m-1inputs with two outputs.For each case,a six-local inequality that the corresponding network necessarily satisfies is obtained and the related quantum violation of this inequality for pure states is investigated,respectively.3.The nonlocality of the tree-shaped quantum network with depth three and containing seven parties and three independent sources is char-acterized,which is applied to detect entanglement of three-partite quan-tum states.Specifically,a three-local inequality that this network nec-essarily satisfies is obtained;it is proved that if all three independent sources produce fully separable quantum states,they definitely do not vi-olate this three-local inequality;when three sources respectively produce two-separable entangled pure states,genuinely entangled pure states and mixed states,there exist such quantum states and quantum measure-ments that violate this three-local inequality;and furthermore,by ex-ploiting quantum violations of this three-local inequality,three schemes are designed to detect entanglement of three-partite quantum states.The related conclusions are respectively generalized to the tree-shaped quan-tum networks with depth three and containing（2ⁿ-1）（n≥3）parties and(2^n-1-1)independent sources and the tree-shaped quantum net-works with depth k（k≥3）and containing（k²-k+1）parties and k independent sources.4.The EPR steering of the two-forked tree-shaped quantum network with depth two and containing seven parties and six independent sources is examined.Specifically,the definition of EPR steering in this network is introduced and a linear steering criterion of whether a network quan-tum state is tree-shaped network steerable is obtained;and based on this,with respect to four classes of important quantum measurements:local orthogonal observables,mutually unbiased measurements,general sym-metric informationally complete positive operator valued measurements and informationally complete（N,M）-positive operator valued measure-ments,specific forms of the steering criterion in the tree-shaped quantum network are respectively derived.