Entanglement and Decoherence in Quantum Walks

量子漫步可以被描述为经典随机漫步的自然对应物，受量子力学原理的支配。近年来，量子游动因其在开发新物种超高效算法方面的潜在应用而成为人们广泛研究的焦点。然而，很大程度上由于涉及的数学挑战，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，特别是在试图联合处理被称为纠缠和退相干的典型量子力学现象的影响方面。通过这个项目，PI的目标是阐明退相干对某些网络中量子行走的影响。他的研究将解决两个问题：1)量子行走的极限分布与量子行走所暴露的退相干程度有何关系？2)子系统(硬币和位置)之间的纠缠程度如何取决于消相干的程度？在这种方法中，纠缠的度量由子系统之间的互信息来反映，而互信息又被定义为冯·诺依曼熵。在量子计算和量子信息领域的研究前沿，这项提议的结果将引起广大活跃调查人员的兴趣。最后但同样重要的是，通过这个项目，Pi的目标是将他的学生，包括研究生和本科生，介绍给新兴的量子信息科学领域。

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.