相干与路径信息

Coherence and path information

近年来,随着对相干性量化的发展,相干与路径信息间的互补关系渐渐引起人们注意.这样的互补关系不仅在基础量子力学方面有重要的理论意义,同时也在量子信息技术中有实际应用.本文从Bures距离和明确量子态区分出发,系统地研究了二路径干涉仪中的相干与路径信息,并建立了一个全新的互补关系.与已知的类似工作相比,本文的研究更为一般:既考虑纯态,也考虑混合态;既探讨了干涉仪本身的路径不对称性所提供的路径可预测性,也探讨了因路径探测器而生的路径可区分度.具体地说,路径可预测性的讨论只依赖于密度矩阵的半正定性质,而路径可区分度的讨论还需要利用保真度和Schur-Weyl不等式等工具.

Complementarity,or more specifically,the wave-particle duality,and quantum coherence are two fundamental concepts in quantum mechanics.Recently,motivated by the progress of the quantification of quantum coherence,the complementary relation between coherence and path information is investigated by many authors,and various duality relations between them are established.Such relations not only provide insights into fundamental problems of quantum mechanics,such as the understanding of quantum coherence and wave-particle duality;but also are important in applications of quantum technologies.In this paper,based on the Bures distance and unambiguous quantum state discrimination,systematic analysis of the complementarity between the quantum coherence and path information in two path interferometers is carried out.Similarly as other related works,the wave aspect,or the visibility of the interferometer,is quantified by the l1-norm measure of quantum coherence,and the path information is considered via unambiguous quantum state discrimination.In this way,a novel duality relation in two path interferometers is obtained.Compared with known results,our work considers mixed states as well as pure states;considers the path predictability resulting from the intrinsic path asymmetry of the quantum state,as well as the path distinguishability resulting from the use of a which-path detector.Therefore,our work systematically generalizes known results in two path interferometers by removing all the unnecessary restrictions.Specifically,the most general form of quantum states in two path interferometers is considered and the duality relation between quantum coherence and path information is proved based on the positivity property of density matrices.The cases of path predictability and path distinguishability are considered separately.For path predictability,the proof is straightforward;whereas some advanced mathematical techniques,such as the Schur-Weyl inequality,properties of the fidelity and properties of positive matrices,are required in order to give a rigorous proof of the duality relation between coherence and path distinguishability.Concrete examples are provided to illustrate the abstract method and results.Our work concerns about two path interferometers exclusively and depends heavily that the dimensionality is two,therefore it would be an interesting task to generalize the results in this paper to n-path interferometers.