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

量子信息处理(QIP)使用量子资源--量子系统、么正演化和测量--来完成信息处理任务。量子现象，如叠加、干涉和纠缠，使使用经典资源难以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更好地理解现有的QIP协议，并有望导致新协议的发展。这一提议通过研究弱测量序列和它们下量子系统的行为来实现这一点。在量子力学中，系统通过两个非常不同的过程在时间上演化：根据薛定谔方程的么正演化和通过测量。么正演化是连续的、可逆的和确定性的；它是量子系统在没有被观察到的情况下经历的演化。相比之下，测量(通常的形式)是不连续的、不可逆转的和随机的。测量提供了有关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的信息量，它必须至少干扰状态一定量。一个较新的概念是弱测量：一种只对系统产生轻微干扰，但只提供非常少量信息的测量。通过重复进行弱测量，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最近表明，任何测量都可以分解为一系列弱测量，其结构具有随机游走的结构：系统的状态随机来回转移到测量的可能结果，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，状态在所有可能状态的空间中沿着一条曲线连续(但随机)扩散。使用这种技术，可以使以前离散的连续过程成为可能。这意味着微积分技术可以应用于量子信息处理中的某些突出问题。一个非常有前景的领域是纠缠。纠缠是一种量子关联，它(在某些方面)比任何经典关联都要强；它是许多量子协议的资源，例如隐形传态和密集编码。出于这个原因，人们对寻找好的纠缠量化方法非常感兴趣。对于一类系统(两部分纯态)，这个问题在很大程度上得到了解决；但对于其他系统，人们知之甚少。事实证明，一个非常有成效的想法是纠缠单调：在纯粹的局部操作下，状态的函数总是平均减小。很难系统地研究这些量；利用弱测量分解，人们可以找到单调的微分条件，并为纠缠问题开辟了一条全新的途径。除了这些测量过程中出现的经典随机行走外，还有纯粹的随机行走的量子类似物，称为量子行走。与随机游走不同，这些是纯粹的么正演化，目前人们非常感兴趣，因为这可能会导致新型量子算法。这个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，以评估这些新算法在现实条件下可能会表现得多好。

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.