Wu Wei, Wang Jin
State Key Laboratory of Electroanalytical Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun 130022, China.
Department of Chemistry, State University of New York at Stony Brook, Stony Brook, NY 11794, USA.
Entropy (Basel). 2020 Jul 24;22(8):813. doi: 10.3390/e22080813.
Wave-particle duality as the defining characteristic of quantum objects is a typical example of the principle of complementarity. The wave-particle-entanglement (WPE) complementarity, initially developed for two-qubit systems, is an extended form of complementarity that combines wave-particle duality with a previously missing ingredient, quantum entanglement. For two-qubit systems in mixed states, the WPE complementarity was further completed by adding yet another piece that characterizes ignorance, forming the wave-particle-entanglement-ignorance (WPEI) complementarity. A general formulation of the WPEI complementarity can not only shed new light on fundamental problems in quantum mechanics, but can also have a wide range of experimental and practical applications in quantum-mechanical settings. The purpose of this study is to establish the WPEI complementarity for general multi-dimensional bipartite systems in pure or mixed states, and extend its range of applications to incorporate hierarchical and infinite-dimensional bipartite systems. The general formulation is facilitated by well-motivated generalizations of the relevant quantities. When faced with different directions of extensions to take, our guiding principle is that the formulated complementarity should be as simple and powerful as possible. We find that the generalized form of the WPEI complementarity contains unequal-weight averages reflecting the difference in the subsystem dimensions, and that the tangle, instead of the squared concurrence, serves as a more suitable entanglement measure in the general scenario. Two examples, a finite-dimensional bipartite system in mixed states and an infinite-dimensional bipartite system in pure states, are studied in detail to illustrate the general formalism. We also discuss our results in connection with some previous work. The WPEI complementarity for general finite-dimensional bipartite systems may be tested in multi-beam interference experiments, while the second example we studied may facilitate future experimental investigations on complementarity in infinite-dimensional bipartite systems.
波粒二象性作为量子物体的决定性特征，是互补原理的一个典型例子。波粒纠缠（WPE）互补性最初是为两量子比特系统发展而来的，它是互补性的一种扩展形式，将波粒二象性与一个先前缺失的要素——量子纠缠结合在一起。对于处于混合态的两量子比特系统，通过添加另一个表征无知的部分，进一步完善了WPE互补性，形成了波粒纠缠无知（WPEI）互补性。WPEI互补性的一般表述不仅能为量子力学中的基本问题带来新的启示，还能在量子力学环境中有广泛的实验和实际应用。本研究的目的是为处于纯态或混合态的一般多维二分系统建立WPEI互补性，并将其应用范围扩展到包含分层和无限维二分系统。通过对相关量进行有充分动机的推广，促进了一般表述的形成。当面临不同的扩展方向时，我们的指导原则是所表述的互补性应尽可能简单且强大。我们发现，WPEI互补性的广义形式包含反映子系统维度差异的不等权重平均值，并且在一般情况下，缠结而非并发平方更适合作为纠缠度量。详细研究了两个例子，一个处于混合态的有限维二分系统和一个处于纯态的无限维二分系统，以说明一般形式体系。我们还结合一些先前的工作讨论了我们的结果。一般有限维二分系统的WPEI互补性可在多光束干涉实验中进行测试，而我们研究的第二个例子可能有助于未来对无限维二分系统中互补性的实验研究。
