Negative Probability and Uncertainty Relations

A concise derivation of all uncertainty relations is given entirely within the context of phase-space quantization, without recourse to operator methods, to the direct use of Weyl's correspondence, or to marginal distributions of x and p.Comment: RevTeX, 4 page

Negative Quasi-Probability as a Resource for Quantum Computation

A central problem in quantum information is to determine the minimal physical resources that are required for quantum computational speedup and, in particular, for fault-tolerant quantum computation. We establish a remarkable connection between the potential for quantum speed-up and the onset of negative values in a distinguished quasi-probability representation, a discrete analog of the Wigner function for quantum systems of odd dimension. This connection allows us to resolve an open question on the existence of bound states for magic-state distillation: we prove that there exist mixed states outside the convex hull of stabilizer states that cannot be distilled to non-stabilizer target states using stabilizer operations. We also provide an efficient simulation protocol for Clifford circuits that extends to a large class of mixed states, including bound universal states.Comment: 15 pages v4: This is a major revision. In particular, we have added a new section detailing an explicit extension of the Gottesman-Knill simulation protocol to deal with positively represented states and measurement (even when these are non-stabilizer). This paper also includes significant elaboration on the two main results of the previous versio

Estimating the probability of large negative stock market

Correct assessment of the risks associated with likely economic outcomes is vital for effective decision making. The objective of investment in the stock market is to obtain positive market returns. The risk, however, is the danger of suffering large negative market returns. A variety of parametric models can be used in assessing this type of risk. A major disadvantage of these techniques is that they require a specific assumption to be made about the nature of the statistical distribution. Projections based on this method are conditional on the validity of this underlying assumption, which itself is not testable. An alternative approach is to use a non-parametric methodology, based on the statistical extreme value theory, which provides a means for evaluating the unconditional distribution (or at least the tails of this distribution) beyond the historically observed values. The methodology involves the calculation of the tail index, which is used to estimate the relevant exceedence probabilities (for different critical levels of loss) for a selection of food industry companies. Information about these downside risks is critically important for investment decision making. In addition, the tail index estimates permit examination of the stable Paretian hypothesis.