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

量子信息处理（QIP）利用量子资源——量子系统、单一演化和测量——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，对测量和纠缠的更好理解，可以期望产生对现有QIP协议的更好理解，并有希望导致新协议的发展。这个提议通过研究弱测量序列和它们下量子系统的行为来解决这个问题。在量子力学中，系统在时间上通过两个非常不同的过程进化：根据薛定谔方程的单一进化，以及通过测量。单一进化是连续的、可逆的和确定的；它是量子系统在不被观察时所经历的进化。相比之下，测量（通常形式）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，它也扰乱了系统的状态。获取信息与干扰系统之间存在着密切的关系；如果测量产生一定数量的信息，它必须至少扰动一定数量的状态。最近的一个想法是弱测量：一种测量只对系统产生轻微的干扰，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最近表明，任何测量都可以分解成一系列弱测量，以一种具有随机游走结构的方式：系统的状态随机地向测量的可能结果来回移动，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，状态在所有可能状态的空间中沿着曲线连续（但随机）扩散。使用这种技术，可以使以前是离散的过程变得连续。这意味着微分学的技术可以用来解决量子信息处理中的一些突出问题。一个非常有前途的领域是纠缠。纠缠是量子相关的一种，在某些方面比任何经典相关都更强；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，人们对寻找纠缠的良好定量测量方法非常感兴趣。这个问题在很大程度上解决了一类系统（二部纯态）；但对其他人来说，我们知之甚少。一个被证明非常富有成效的想法是纠缠单调：在纯局部操作下，状态的函数平均总是减少。系统地研究这些数量是很困难的；利用弱测量分解，可以找到单调的微分条件，为解决纠缠问题开辟了一条全新的途径。除了在这些测量过程中发生的经典随机漫步之外，还有随机漫步的纯量子类似物，称为量子漫步。与随机漫步不同，这些是纯粹的一元进化，这是目前非常感兴趣的，可能会导致新型量子算法。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预期性能。

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.