Analysis of Absorbing Times of Quantum Walks

Quantum walks are expected to provide useful algorithmic tools for quantum computation. This paper introduces absorbing probability and time of quantum walks and gives both numerical simulation results and theoretical analyses on Hadamard walks on the line and symmetric walks on the hypercube from the viewpoint of absorbing probability and time.Comment: LaTeX2e, 14 pages, 6 figures, 1 table, figures revised, references added, to appear in Physical Review

Response probability and latency: a straight line, an operational definition of meaning and the structure of short term memory

The functional relationship between response probability and time is investigated in data from Rubin, Hinton and Wenzel (1999) and Anderson (1981). Recall/recognition probabilities and search times are linearly related through stimulus presentation lags from 6 seconds to 600 seconds in the former experiment and for repeated learning of words in the latter. The slope of the response time vs. probability function is related to the meaningfulness of the items used. The Rubin et al data suggest that only one memory structure is present or that all memory structures probed show the same linear relation of response probability and time. Both sets of data also suggest that the memory items, presumably in the neocortex, have a finite effective size that shrinks in a logarithmic fashion as the time since stimulus presentation increases or the overlearning decreases, away from the start of the search. According to the logarithmic decay, the size of the memory items decreases to a couple of neurons at about 1500 seconds for recall and 1100 seconds for recognition – this could be the time scale for a short term memory being converted to a long term memory. The incorrect recall time saturates in the Rubin et al data (it is not linear throughout the experiments), suggesting a limited size of the short term memory structure: the time to search through the structure for recall is 1.7 seconds. For recognition the corresponding time is about 0.4 seconds, to compare with the 0.243 seconds given by the data analysis of Cavanagh of Sternberg-like experiments (1972)

Philosophical lessons of entanglement

The quantum-mechanical description of the world, including human observers, makes substantial use of entanglement. In order to understand this, we need to adopt concepts of truth, probability and time which are unfamiliar in modern scientific thought. There are two kinds of statements about the world: those made from inside the world, and those from outside. The conflict between contradictory statements which both appear to be true can be resolved by recognising that they are made in different perspectives. Probability, in an objective sense, belongs in the internal perspective, and to statements in the future tense. Such statements obey a many-valued logic, in which the truth values are identified as probabilities.Comment: Talk given at 75 Years of Quantum Entanglement, Kolkata, India, 10 January 201

Weyl's principle, cosmic time and quantum fundamentalism

We examine the necessary physical underpinnings for setting up the cosmological standard model with a global cosmic time parameter. In particular, we discuss the role of Weyl's principle which asserts that cosmic matter moves according to certain regularity requirements. After a brief historical introduction to Weyl's principle we argue that although the principle is often not explicitly mentioned in modern standard texts on cosmology, it is implicitly assumed and is, in fact, necessary for a physically well-defined notion of cosmic time. We finally point out that Weyl's principle might be in conflict with the wide-spread idea that the universe at some very early stage can be described exclusively in terms of quantum theory.Comment: To appear in the section on "Physical and philosophical perspectives on probability and time" in S. Hartmann et al. (eds.) "Explanation, Prediction and Confirmation", Springer's The Philosophy of Science in a European Perspective book serie

Modeling and Analysis of Probabilistic Real-time Systems through Integrating Event-B and Probabilistic Model Checking

Event-B is a formal method used in the development of safety critical systems. However, these systems may introduce uncertainty, and need also to meet real-time requirements, which make their modeling and analysis a challenging task. Existing works on extending Event-B with probability and time did not address both probability and time in a single framework. Besides, they did focus the most on extending the language itself, not on integrating the extended Event-B with verification. In this paper, we aim to represent both probability and time in the Event-B language, and we will show how such a representation can be automatically translated into Probabilistic Timed Automata (PTA) described in the language of the probabilistic model checker PRISM. This translation would allow us to analyze probabilistic, as well as time-bounded probabilistic reachability properties of probabilistic real-time systems through the Probabilistic Timed CTL (PTCTL) logic

Quantile calculus and censored regression

Quantile regression has been advocated in survival analysis to assess evolving covariate effects. However, challenges arise when the censoring time is not always observed and may be covariate-dependent, particularly in the presence of continuously-distributed covariates. In spite of several recent advances, existing methods either involve algorithmic complications or impose a probability grid. The former leads to difficulties in the implementation and asymptotics, whereas the latter introduces undesirable grid dependence. To resolve these issues, we develop fundamental and general quantile calculus on cumulative probability scale in this article, upon recognizing that probability and time scales do not always have a one-to-one mapping given a survival distribution. These results give rise to a novel estimation procedure for censored quantile regression, based on estimating integral equations. A numerically reliable and efficient Progressive Localized Minimization (PLMIN) algorithm is proposed for the computation. This procedure reduces exactly to the Kaplan--Meier method in the $k$-sample problem, and to standard uncensored quantile regression in the absence of censoring. Under regularity conditions, the proposed quantile coefficient estimator is uniformly consistent and converges weakly to a Gaussian process. Simulations show good statistical and algorithmic performance. The proposal is illustrated in the application to a clinical study.Comment: Published in at http://dx.doi.org/10.1214/09-AOS771 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org