加权导引不等式的构造与应用问题研究

Quantum steering is a type of quantum nonlocality weaker than Bell nonlocality. In this project, by using operator theory and spectral theory and combining numerical simulation and analysis, the construction and application of weighted steering inequalities are investigated. The main research contents and objectives are as follows: (a) based on weighted Bell inequalities on bi-partite system, the general method of constructing weighted steering inequalities is presented, the observables capable of constructing weighted steering inequalities are optimized in the sense of detecting more steerable isotropic states, which yields that the optimal weighted steering inequality is obtained, and then the steerable state are characterized; (b) using these new weighted steering inequalities, the relationship between the maximal violation of steering inequalities and the linear entropy of states is established, and the states with maximal violation of steering inequalities and maximal mixedness are investigated; (c) on qudit-qudit(d>2) systems, the universal method of constructing Bell nonlocal states by steering states is studied, the quantitative relationship between quantum steering and Bell nonlocality is established, based on this quantitative relationship Bell nonlocality can be detected by the violation of the new weighted steering inequalities, and the experimental verification of this quantitative relationship is presented. The expected results can provide theoretical foundation for quantum computation and quantum information processing tasks such as quantum key distribution, quantum cryptography and so on.

量子导引是比Bell非定域性更弱的一种量子非定域性。本项目应用算子理论与谱理论，结合数值模拟与分析，研究加权导引不等式的构造与应用问题，研究内容与目标为：基于两体系统上的加权Bell不等式，给出构造加权导引不等式的一般方法，在探测更多可导引isotropic态的意义下，优化可用于构造加权导引不等式的可观测量，得到最优加权导引不等式，刻画可导引量子态的结构；利用新构造的加权导引不等式，建立量子态的导引不等式极大破坏与线性熵之间的联系，刻画极大可导引极大混合量子态的结构；研究两体高维系统上由可导引态构造Bell非定域态的普适方法，建立两体高维系统上量子导引与Bell非定域性之间的定量关系，利用此定量关系与新构造的加权导引不等式的违背，探测Bell非定域性，并给出此定量关系验证的实验方案。预期成果将为量子秘钥分发、量子加密等量子信息处理任务奠定理论基础。

本项目研究加权导引不等式的构造与应用问题，贝尔不等式是构造加权导引不等式的基础，稳健贝尔不等式是构造最优加权导引不等式的难点。在稳健贝尔不等式研究过程中我们发现多体三维系统上稳健贝尔不等式的迭代规律，研究成果发表在国际主流期刊Physical Review A上，这不仅是稳健贝尔不等式研究的重要进展，也对研究最优加权导引不等式奠定基础。本项目还探索了多体任意维系统上稳健贝尔不等式的迭代规律，研究目标是将已知的稳健贝尔不等式统一起来，给出一般的多体任意维系统上稳健贝尔不等式的迭代规律，完善利用不等式研究贝尔非定域性的框架。量子互文是比贝尔非定域性更一般的量子关联，本项目也对量子互文展开了深入的调查，给出了成功概率更高的量子互文Hardy-like 证明，此成果还提出了7测量4维系统上量子互文Hardy-like 证明成功概率验证的实验方案，实验结果在误差允许范围内与理论结果一致，研究成果发表在中科院一区Photonics Research 杂志。