De-Quantising the Solution of Deutsch's Problem

Probably the simplest and most frequently used way to illustrate the power of quantum computing is to solve the so-called {\it Deutsch's problem}. Consider a Boolean function $f: \{0,1\} \to \{0,1\}$ and suppose that we have a (classical) black box to compute it. The problem asks whether $f$ is constant (that is, $f(0) = f(1)$) or balanced ($f(0) \not= f(1)$). Classically, to solve the problem seems to require the computation of $f(0)$ and $f(1)$, and then the comparison of results. Is it possible to solve the problem with {\em only one} query on $f$? In a famous paper published in 1985, Deutsch posed the problem and obtained a ``quantum'' {\em partial affirmative answer}. In 1998 a complete, probability-one solution was presented by Cleve, Ekert, Macchiavello, and Mosca. Here we will show that the quantum solution can be {\it de-quantised} to a deterministic simpler solution which is as efficient as the quantum one. The use of ``superposition'', a key ingredient of quantum algorithm, is--in this specific case--classically available.Comment: 8 page

Persistent accelerations disentangle Lagrangian turbulence

Particles in turbulence frequently encounter extreme accelerations between extended periods of quiescence. The occurrence of extreme events is closely related to the intermittent spatial distribution of intense flow structures such as vorticity filaments. This mixed history of flow conditions leads to very complex particle statistics with a pronounced scale dependence, which presents one of the major challenges on the way to a non-equilibrium statistical mechanics of turbulence. Here, we introduce the notion of persistent Lagrangian acceleration, quantified by the squared particle acceleration coarse-grained over a viscous time scale. Conditioning Lagrangian particle data from simulations on this coarse-grained acceleration, we find remarkably simple, close-to-Gaussian statistics for a range of Reynolds numbers. This opens the possibility to decompose the complex particle statistics into much simpler sub-ensembles. Based on this observation, we develop a comprehensive theoretical framework for Lagrangian single-particle statistics that captures the acceleration, velocity increments as well as single-particle dispersion

On the Thermodynamics of NUT charged spaces

We discuss and compare at length the results of two methods used recently to describe the thermodynamics of Taub-NUT solutions in a deSitter background. In the first approach (\mathbb{% C}-approach), one deals with an analytically continued version of the metric while in the second approach ($\mathbb{R}$-approach), the discussion is carried out using the unmodified metric with Lorentzian signature. No analytic continuation is performed on the coordinates and/or the parameters that appear in the metric. We find that the results of both these approaches are completely equivalent modulo analytic continuation and we provide the exact prescription that relates the results in both methods. The extension of these results to the AdS/flat cases aims to give a physical interpretation of the thermodynamics of nut-charged spacetimes in the Lorentzian sector. We also briefly discuss the higher dimensional spaces and note that, analogous with the absence of hyperbolic nuts in AdS backgrounds, there are no spherical Taub-Nut-dS solutions.Comment: 35pages, 4 figures. v.4 references added,few typos corrected, to appear in Phys. Rev.

Natural Halting Probabilities, Partial Randomness, and Zeta Functions

We introduce the zeta number, natural halting probability and natural complexity of a Turing machine and we relate them to Chaitin's Omega number, halting probability, and program-size complexity. A classification of Turing machines according to their zeta numbers is proposed: divergent, convergent and tuatara. We prove the existence of universal convergent and tuatara machines. Various results on (algorithmic) randomness and partial randomness are proved. For example, we show that the zeta number of a universal tuatara machine is c.e. and random. A new type of partial randomness, asymptotic randomness, is introduced. Finally we show that in contrast to classical (algorithmic) randomness--which cannot be naturally characterised in terms of plain complexity--asymptotic randomness admits such a characterisation.Comment: Accepted for publication in Information and Computin

Massive stealth scalar fields from field redefinition method

We propose an uni-parametric deformation method of action principles of scalar fields coupled to gravity which generates new models with massive stealth field configurations, i.e. with vanishing energy-momentum tensor. The method applies to a wide class of models and we provide three examples. In particular we observe that in the case of the standard massive scalar action principle, the respective deformed action contains the stealth configurations and it preserves the massive ones of the undeformed model. We also observe that, in this latter example, the effect of the energy-momentum tensor of the massive (non-stealth) field can be amplified or damped by the deformation parameter, alternatively the mass of the stealth field.Comment: 12 page