Quantum Query Algorithms are Completely Bounded Forms

We prove a characterization of $t$-query quantum algorithms in terms of the unit ball of a space of degree-$2t$ polynomials. Based on this, we obtain a refined notion of approximate polynomial degree that equals the quantum query complexity, answering a question of Aaronson et al. (CCC'16). Our proof is based on a fundamental result of Christensen and Sinclair (J. Funct. Anal., 1987) that generalizes the well-known Stinespring representation for quantum channels to multilinear forms. Using our characterization, we show that many polynomials of degree four are far from those coming from two-query quantum algorithms. We also give a simple and short proof of one of the results of Aaronson et al. showing an equivalence between one-query quantum algorithms and bounded quadratic polynomials.Comment: 24 pages, 3 figures. v2: 27 pages, minor changes in response to referee comment

Locally Complete Path Independent Choice Functions and Their Lattices

The concept of path independence (PI) was first introduced by Arrow (1963) as a defense of his requirement that collective choices be rationalized by a weak ordering. Plott (1973) highlighted the dynamic aspects of PI implicit in Arrow's initial discussion. Throughout these investigations two questions, both initially raised by Plott, remained unanswered. What are the precise mathematical foundations for path independence? How can PI choice functions be constructed? We give complete answers to both these questions for finite domains and provide necessary conditions for infinite domains. We introduce a lattice associated with each PI function. For finite domains these lattices coincide with locally lower distributive or meet-distributive lattices and uniquely characterize PI functions. We also present an algorithm, effective and exhaustive for finite domains, for the construction of PI choice functions and hence for all finite locally lower distributive lattices. For finite domains, a PI function is rationalizable if and only if the lattice is distributive. The lattices associated with PI functions that satisfy the stronger condition of the weak axiom of revealed preference are chains of Boolean algebras and conversely. Those that satisfy the strong axiom of revealed preference are chains and conversely.

A unique factorization theorem for matroids

We study the combinatorial, algebraic and geometric properties of the free product operation on matroids. After giving cryptomorphic definitions of free product in terms of independent sets, bases, circuits, closure, flats and rank function, we show that free product, which is a noncommutative operation, is associative and respects matroid duality. The free product of matroids $M$ and $N$ is maximal with respect to the weak order among matroids having $M$ as a submatroid, with complementary contraction equal to $N$. Any minor of the free product of $M$ and $N$ is a free product of a repeated truncation of the corresponding minor of $M$ with a repeated Higgs lift of the corresponding minor of $N$. We characterize, in terms of their cyclic flats, matroids that are irreducible with respect to free product, and prove that the factorization of a matroid into a free product of irreducibles is unique up to isomorphism. We use these results to determine, for K a field of characteristic zero, the structure of the minor coalgebra $\cal C$ of a family of matroids $\cal M$ that is closed under formation of minors and free products: namely, $\cal C$ is cofree, cogenerated by the set of irreducible matroids belonging to $\cal M$.Comment: Dedicated to Denis Higgs. 25 pages, 3 figures. Submitted for publication in the Journal of Combinatorial Theory (A). See arXiv:math.CO/0409028 arXiv:math.CO/0409080 for preparatory work on this subjec