QnTM: Weak Local Measurements, Entanglement Monotones, and Random Walks

QnTM：弱局部测量、纠缠单调和随机游走

• 批准号: 0524822
• 负责人: Todd Brun
• 金额: $ 15万
• 依托单位: University of Southern California
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-08-01 至 2008-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0524822&HistoricalAwards=false
• 关键词: QnTM; Weak; Local; Measurements; Entanglement

Quantum information processing (QIP) uses quantum resources-quantum systems, unitary evolutions, and measurement-to do information processing tasks. Quantum phenomena, such as superposition, interference, and entanglement, make possible protocols that are difficult or impossible using classical resources. A better understanding of measurement and entanglement, therefore, can be expected to yield a better understanding of existing QIP protocols, and hopefully lead to the development of new protocols as well. This proposal approaches this by studying sequences of weak measurements, and the behavior of quantum systems under them.In quantum mechanics, systems evolve in time by two very different processes: by unitary evolution according to the Schrdinger equation, and by measurement. Unitary evolution is continuous, reversible, and deterministic; it is the evolution that quantum systems undergo when they are not observed. By contrast, measurement (in its usual form) is discontinuous, irreversible, and random. A measurement provides some information about the state of a quantum system; but at the same time, it disturbs the state of the system. There is a close relationship between acquiring information and disturbing the system; if a measurement yields a certain amount of information, it must disturb the state by at least a certain amount.A more recent idea is that of a weak measurement: a measurement that disturbs the system only slightly, but provides only a very small amount of information. By repeatedly doing weak measurements, more and more information can be accumulated (and the disturbance grows progressively greater and greater). In fact, the PI has recently shown that any measurement can be decomposed into a sequence of weak measurements, in a way that has the structure of a random walk: the state of the system shifts randomly back and forth towards the possible outcomes of the measurement, and at long times is guaranteed to approach one or another of the outcomes with a given probability. In the limit, this is like a diffusion process, with the state diffusing continuously (but randomly) along a curve in the space of all possible states.Using this technique, it is possible to make continuous processes that previously were discrete. This means that the techniques of differential calculus can be brought to bear on certain outstanding problems in quantum information processing. One very promising area is entanglement. Entanglement is a type of quantum correlation, which is stronger (in certain ways) than any classical correlation; it is a resource for a number of quantum protocols, such as teleportation and dense coding. For this reason, there has been a great deal of interest in finding good quantitative measures of entanglement. This problem is largely solved for one class of systems (bipartite pure states); but for others, little is known. An idea that has proven very fruitful is that of an entanglement monotone: a function of the state that always decreases on average under purely local operations. It has been difficult to investigate these quantities systematically; using weak measurement decompositions, one can find differential conditions for monotones, and open up a brand new avenue to the problem of entanglement.In addition to the classical random walks that occur in these measurement procedures, there are purely quantum analogues of random walks, called quantum walks. Unlike the random walks, these are purely unitary evolutions, which are currently of great interest as possibly leading to new types of quantum algorithms. This project will also study quantum walks on graphs, with particular emphasis on the effects of decoherence (quantum noise) and other imperfections, to assess how well such new algorithms might be expected to perform under realistic conditions.

量子信息处理（QIP）使用量子资源-量子系统，幺正演化和测量-来完成信息处理任务。 量子现象，如叠加、干涉和纠缠，使得使用经典资源很难或不可能实现的协议成为可能。 因此，更好地理解测量和纠缠，可以预期产生更好地理解现有的QIP协议，并有希望导致新协议的发展。 在量子力学中，系统在时间上通过两种完全不同的过程演化：根据薛定谔方程的幺正演化和通过测量演化。 幺正演化是连续的、可逆的和确定性的;它是量子系统在未被观测到时所经历的演化。 相比之下，测量（通常的形式）是不连续的、不可逆的和随机的。 测量提供了关于量子系统状态的一些信息;但同时，它也干扰了系统的状态。 获取信息和干扰系统之间有着密切的关系;如果一个测量产生了一定量的信息，它必须至少以一定的量干扰状态。最近的一个想法是弱测量：一个测量只轻微地干扰系统，但只提供非常少量的信息。 通过反复进行弱测量，可以积累越来越多的信息（并且干扰逐渐变得越来越大）。 事实上，PI最近已经表明，任何测量都可以分解为一系列弱测量，以具有随机游走结构的方式：系统的状态朝着测量的可能结果随机来回移动，并且在很长一段时间内保证以给定的概率接近一个或另一个结果。 在极限情况下，这就像一个扩散过程，状态沿着一条曲线在所有可能状态的空间中连续扩散（但随机）。 这意味着微分学技术可以用来解决量子信息处理中的某些突出问题。 一个非常有前途的领域是纠缠。 纠缠是一种量子关联，在某些方面比任何经典关联都强;它是许多量子协议的资源，例如隐形传态和密集编码。 由于这个原因，人们对找到好的量子纠缠度有很大的兴趣。 这个问题在很大程度上解决了一类系统（二分纯态），但对其他人来说，知之甚少。 一个被证明非常富有成效的想法是纠缠单调性：一个在纯局部操作下总是平均减少的状态函数。 系统地研究这些物理量是很困难的，但利用弱测量分解，我们可以找到单调性的微分条件，从而为纠缠问题开辟了一条全新的途径。除了在这些测量过程中出现的经典随机游走之外，还有随机游走的纯粹量子类似物，称为量子游走。 与随机游走不同，这些是纯粹的酉演化，目前人们对它非常感兴趣，因为它可能导致新类型的量子算法。 该项目还将研究图上的量子行走，特别强调退相干（量子噪声）和其他缺陷的影响，以评估这种新算法在现实条件下的表现。