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

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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.