We give a useful new characterization of the set of all completely positive, trace-preserving maps Φ : M2 → M2 from which one can easily check any trace-preserving map for complete positivity. We… (More)

For a (compact) subset K of a metric space and ε > 0, the covering number N(K, ε) is defined as the smallest number of balls of radius ε whose union covers K. Knowledge of the metric entropy, i.e.,… (More)

In this note we give sharp estimates on the volume of the set of separable states on N qubits. In particular, the magnitude of the “effective radius” of that set in the sense of volume is determined… (More)

For a subset I of [1, ..., n], let us denote by PI the natural projection from [0, 1] to [0, 1]. The Sauer Shelah lemma [Sa], [Sh] asserts that given a subset A of [0, 1] and an integer k with card… (More)

For two convex bodies K and T in Rn, the covering number of K by T , denoted N(K, T ), is defined as the minimal number of translates of T needed to cover K. Let us denote by K◦ the polar body of K… (More)

Let L be a lattice in IRn and K a convex body disjoint from L. The classical Flatness Theorem asserts that then w(K, L), the L-width of K, doesn’t exceed some bound, depending only on the dimension… (More)

Let μ be a Gaussian measure (say, on R) and let K, L ⊆ R be such that K is convex, L is a “layer” (i.e. L = {x : a ≤ 〈x, u〉 ≤ b} for some a, b ∈ R and u ∈ R) and the centers of mass (with respect to… (More)

One of the major challenges for collective intelligence is inconsistency, which is unavoidable whenever subjective assessments are involved. Pairwise comparisons allow one to represent such… (More)