Entanglement and Decoherence in Quantum Walks

量子游走可以被描述为经典随机游走的自然对应物，由量子力学原理支配。 近年来，由于量子漫步在开发一种新的超高效算法方面的潜在应用，量子漫步已经成为广泛研究兴趣的焦点。 然而，很大程度上由于所涉及的数学挑战，该理论的一些重要方面仍然相对未被探索。 这些挑战一直存在，特别是在试图联合治疗的影响，被称为“纠缠”和“退相干”的量子力学现象的特征。通过这个项目，PI的目的是阐明退相干的量子行走在某些网络的影响。 他的研究将解决两个问题：1）量子游走的极限分布与量子游走所暴露的退相干水平有何关系？2）子系统（硬币和位置）之间的纠缠度如何依赖于退相干的水平？ 在这种方法中，纠缠的度量由子系统之间的互信息来反映，而互信息又由冯诺依曼熵来定义。这一提议的结果将引起量子计算和量子信息领域研究前沿的广大活跃研究人员的兴趣。 最后但并非最不重要的是，通过这个项目，PI的目标是向他的学生（包括研究生和本科生）介绍量子信息科学的新兴领域。

Quantum walks may be described as the natural counterparts of classical random walks, governed by the principles of quantum mechanics. In recent times, because of their potential applications to the development of a new species of super-efficient algorithms, quantum walks have been the focus of extensive research interest. However, largely due to the mathematical challenges involved, some important aspects of the theory have remained relatively unexplored. These challenges have persisted, especially in connection with attempts to treat jointly the effects of the characteristically quantum-mechanical phenomena known as "entanglement" and "decoherence".Through this project, PI aims to elucidate the effects of decoherence on quantum walks over certain networks. His study will address two questions: 1) How is the limiting distribution of the quantum walk related to the level of decoherence to which the quantum walk is exposed? and 2) How does the measure of entanglement between the subsystems (coin and position) depend upon the level of decoherence? In this approach, the measure of entanglement is reflected by the mutual information between the subsystems, which, in turn, is defined in terms of the von Neumann entropy. The outcomes to flow from this proposal will be of interest to a wide audience of active investigators at the forefront of research in the fields of quantum computing and quantum information. Last but not least, through this project, PI aims to introduce his students, both graduate and undergraduate, to the emerging field of Quantum Information Science.