Optimal randomness generation from optical Bell experiments

Genuine randomness can be certified from Bell tests without any detailed assumptions on the working of the devices with which the test is implemented. An important class of experiments for implementing such tests is optical setups based on polarisation measurements of entangled photons distributed from a spontaneous parametric down conversion source. Here we compute the maximal amount of randomness which can be certified in such setups under realistic conditions. We provide relevant yet unexpected numerical values for the physical parameters and achieve four times more randomness than previous methods.Comment: 15 pages, 4 figure

Uniform test of algorithmic randomness over a general space

The algorithmic theory of randomness is well developed when the underlying space is the set of finite or infinite sequences and the underlying probability distribution is the uniform distribution or a computable distribution. These restrictions seem artificial. Some progress has been made to extend the theory to arbitrary Bernoulli distributions (by Martin-Loef), and to arbitrary distributions (by Levin). We recall the main ideas and problems of Levin's theory, and report further progress in the same framework. - We allow non-compact spaces (like the space of continuous functions, underlying the Brownian motion). - The uniform test (deficiency of randomness) d_P(x) (depending both on the outcome x and the measure P should be defined in a general and natural way. - We see which of the old results survive: existence of universal tests, conservation of randomness, expression of tests in terms of description complexity, existence of a universal measure, expression of mutual information as "deficiency of independence. - The negative of the new randomness test is shown to be a generalization of complexity in continuous spaces; we show that the addition theorem survives. The paper's main contribution is introducing an appropriate framework for studying these questions and related ones (like statistics for a general family of distributions).Comment: 40 pages. Journal reference and a slight correction in the proof of Theorem 7 adde

Simulation-based Tests that Can Use Any Number of Simulations

Conventional procedures for Monte Carlo and bootstrap tests require that B, the number of simulations, satisfy a specific relationship with the level of the test. Otherwise, a test that would instead be exact will either overreject or underreject for finite B. We present expressions for the rejection frequencies associated with existing procedures and propose a new procedure that yields exact Monte Carlo tests for any positive value of B. This procedure, which can also be used for bootstrap tests, is likely to be most useful when simulation is expensive.resampling, Monte Carlo test, bootstrap test, percentiles, simulation

Particle-hole symmetric localization in two dimensions

We revisit two-dimensional particle-hole symmetric sublattice localization problem, focusing on the origin of the observed singularities in the density of states $\rho(E)$ at the band center E=0. The most general such system [R. Gade, Nucl. Phys. B {\bf 398}, 499 (1993)] exhibits critical behavior and has $\rho(E)$ that diverges stronger than any integrable power-law, while the special {\it random vector potential model} of Ludwiget al [Phys. Rev. B {\bf 50}, 7526 (1994)] has instead a power-law density of states with a continuously varying dynamical exponent. We show that the latter model undergoes a dynamical transition with increasing disorder--this transition is a counterpart of the static transition known to occur in this system; in the strong-disorder regime, we identify the low-energy states of this model with the local extrema of the defining two-dimensional Gaussian random surface. Furthermore, combining this ``surface fluctuation'' mechanism with a renormalization group treatment of a related vortex glass problem leads us to argue that the asymptotic low $E$ behavior of the density of states in the {\it general} case is $\rho(E) \sim E^{-1} e^{-|\ln E|^{2/3}}$, different from earlier prediction of Gade. We also study the localized phases of such particle-hole symmetric systems and identify a Griffiths ``string'' mechanism that generates singular power-law contributions to the low-energy density of states in this case.Comment: 18 pages (two-column PRB format), 10 eps figures include

Stochastic kinetics of ribosomes: single motor properties and collective behavior

Synthesis of protein molecules in a cell are carried out by ribosomes. A ribosome can be regarded as a molecular motor which utilizes the input chemical energy to move on a messenger RNA (mRNA) track that also serves as a template for the polymerization of the corresponding protein. The forward movement, however, is characterized by an alternating sequence of translocation and pause. Using a quantitative model, which captures the mechanochemical cycle of an individual ribosome, we derive an {\it exact} analytical expression for the distribution of its dwell times at the successive positions on the mRNA track. Inverse of the average dwell time satisfies a ``Michaelis-Menten-like'' equation and is consistent with the general formula for the average velocity of a molecular motor with an unbranched mechano-chemical cycle. Extending this formula appropriately, we also derive the exact force-velocity relation for a ribosome. Often many ribosomes simultaneously move on the same mRNA track, while each synthesizes a copy of the same protein. We extend the model of a single ribosome by incorporating steric exclusion of different individuals on the same track. We draw the phase diagram of this model of ribosome traffic in 3-dimensional spaces spanned by experimentally controllable parameters. We suggest new experimental tests of our theoretical predictions.Comment: Final published versio

Derandomization and Group Testing

The rapid development of derandomization theory, which is a fundamental area in theoretical computer science, has recently led to many surprising applications outside its initial intention. We will review some recent such developments related to combinatorial group testing. In its most basic setting, the aim of group testing is to identify a set of "positive" individuals in a population of items by taking groups of items and asking whether there is a positive in each group. In particular, we will discuss explicit constructions of optimal or nearly-optimal group testing schemes using "randomness-conducting" functions. Among such developments are constructions of error-correcting group testing schemes using randomness extractors and condensers, as well as threshold group testing schemes from lossless condensers.Comment: Invited Paper in Proceedings of 48th Annual Allerton Conference on Communication, Control, and Computing, 201