Approximate Degree, Secret Sharing, and Concentration Phenomena

The epsilon-approximate degree deg~_epsilon(f) of a Boolean function f is the least degree of a real-valued polynomial that approximates f pointwise to within epsilon. A sound and complete certificate for approximate degree being at least k is a pair of probability distributions, also known as a dual polynomial, that are perfectly k-wise indistinguishable, but are distinguishable by f with advantage 1 - epsilon. Our contributions are: - We give a simple, explicit new construction of a dual polynomial for the AND function on n bits, certifying that its epsilon-approximate degree is Omega (sqrt{n log 1/epsilon}). This construction is the first to extend to the notion of weighted degree, and yields the first explicit certificate that the 1/3-approximate degree of any (possibly unbalanced) read-once DNF is Omega(sqrt{n}). It draws a novel connection between the approximate degree of AND and anti-concentration of the Binomial distribution. - We show that any pair of symmetric distributions on n-bit strings that are perfectly k-wise indistinguishable are also statistically K-wise indistinguishable with at most K^{3/2} * exp (-Omega (k^2/K)) error for all k < K <= n/64. This bound is essentially tight, and implies that any symmetric function f is a reconstruction function with constant advantage for a ramp secret sharing scheme that is secure against size-K coalitions with statistical error K^{3/2} * exp (-Omega (deg~_{1/3}(f)^2/K)) for all values of K up to n/64 simultaneously. Previous secret sharing schemes required that K be determined in advance, and only worked for f=AND. Our analysis draws another new connection between approximate degree and concentration phenomena. As a corollary of this result, we show that for any d deg~_{1/3}(f). These upper and lower bounds were also previously only known in the case f=AND

Optimal Geo-Indistinguishable Mechanisms for Location Privacy

We consider the geo-indistinguishability approach to location privacy, and the trade-off with respect to utility. We show that, given a desired degree of geo-indistinguishability, it is possible to construct a mechanism that minimizes the service quality loss, using linear programming techniques. In addition we show that, under certain conditions, such mechanism also provides optimal privacy in the sense of Shokri et al. Furthermore, we propose a method to reduce the number of constraints of the linear program from cubic to quadratic, maintaining the privacy guarantees and without affecting significantly the utility of the generated mechanism. This reduces considerably the time required to solve the linear program, thus enlarging significantly the location sets for which the optimal mechanisms can be computed.Comment: 13 page

Overcoming phonon-induced dephasing for indistinguishable photon sources

Reliable single photon sources constitute the basis of schemes for quantum communication and measurement based quantum computing. Solid state single photon sources based on quantum dots are convenient and versatile but the electronic transitions that generate the photons are subject to interactions with lattice vibrations. Using a microscopic model of electron-phonon interactions and a quantum master equation, we here examine phonon-induced decoherence and assess its impact on the rate of production, and indistinguishability, of single photons emitted from an optically driven quantum dot system. We find that, above a certain threshold of desired indistinguishability, it is possible to mitigate the deleterious effects of phonons by exploiting a three-level Raman process for photon production

Sharp Indistinguishability Bounds from Non-Uniform Approximations

We study the basic problem of distinguishing between two symmetric probability distributions over n bits by observing k bits of a sample, subject to the constraint that all (k-1)-wise marginal distributions of the two distributions are identical to each other. Previous works of Bogdanov et al. [Bogdanov et al., 2019] and of Huang and Viola [Huang and Viola, 2019] have established approximately tight results on the maximal possible statistical distance between the k-wise marginals of such distributions when k is at most a small constant fraction of n. Naor and Shamir [Naor and Shamir, 1994] gave a tight bound for all k in the special case k = n and when distinguishing with the OR function; they also derived a non-tight result for general k and n. Krause and Simon [Krause and Simon, 2000] gave improved upper and lower bounds for general k and n when distinguishing with the OR function, but these bounds are exponentially far apart when k = ?(n). In this work we provide sharp upper and lower bounds on the maximal statistical distance that hold for all k and n. Upper bounds on the statistical distance have typically been obtained by providing uniform low-degree polynomial approximations to certain higher-degree polynomials. This is the first work to construct suitable non-uniform approximations for this purpose; the sharpness and wider applicability of our result stems from this non-uniformity

Nose-Hoover sampling of quantum entangled distribution functions

While thermostated time evolutions stand on firm grounds and are widely used in classical molecular dynamics (MD) simulations, similar methods for quantum MD schemes are still lacking. In the special case of a quantum particle in a harmonic potential, it has been shown that the framework of coherent states permits to set up equations of motion for an isothermal quantum dynamics. In the present article, these results are generalized to indistinguishable quantum particles. We investigate the consequences of the (anti-)symmetry of the many-particle wavefunction which leads to quantum entangled distribution functions. The resulting isothermal equations of motion for bosons and fermions contain new terms which cause Bose-attraction and Pauli-blocking. Questions of ergodicity are discussed for different coupling schemes.Comment: 15 pages, 4 figures, submitted to PHYSICA A. More information at http://www.physik.uni-osnabrueck.de/makrosysteme

Computing a T-transitive lower approximation or opening of a proximity relation

Fuzzy Sets and Systems. IMPACT FACTOR: 1,181. Fuzzy Sets and Systems. IMPACT FACTOR: 1,181. Since transitivity is quite often violated even by decision makers that accept transitivity in their preferences as a condition for consistency, a standard approach to deal with intransitive preference elicitations is the search for a close enough transitive preference relation, assuming that such a violation is mainly due to decision maker estimation errors. In some way, the more number of elicitations, the more probable inconsistency is. This is mostly the case within a fuzzy framework, even when the number of alternatives or object to be classified is relatively small. In this paper we propose a fast method to compute a T-indistinguishability from a reflexive and symmetric fuzzy relation, being T any left-continuous t-norm. The computed approximation we propose will take O(n3) time complexity, where n is the number of elements under consideration, and is expected to produce a T-transitive opening. To the authors¿ knowledge, there are no other proposed algorithm that computes T-transitive lower approximations or openings while preserving the reflexivity and symmetry properties