On the Computational Complexity of Defining Sets

Suppose we have a family ${\cal F}$ of sets. For every $S \in {\cal F}$, a set $D \subseteq S$ is a {\sf defining set} for $({\cal F},S)$ if $S$ is the only element of $\cal{F}$ that contains $D$ as a subset. This concept has been studied in numerous cases, such as vertex colorings, perfect matchings, dominating sets, block designs, geodetics, orientations, and Latin squares. In this paper, first, we propose the concept of a defining set of a logical formula, and we prove that the computational complexity of such a problem is $\Sigma_2$-complete. We also show that the computational complexity of the following problem about the defining set of vertex colorings of graphs is $\Sigma_2$-complete: {\sc Instance:} A graph $G$ with a vertex coloring $c$ and an integer $k$. {\sc Question:} If ${\cal C}(G)$ be the set of all $\chi(G)$-colorings of $G$, then does $({\cal C}(G),c)$ have a defining set of size at most $k$? Moreover, we study the computational complexity of some other variants of this problem

Weighted complex projective 2-designs from bases: optimal state determination by orthogonal measurements

We introduce the problem of constructing weighted complex projective 2-designs from the union of a family of orthonormal bases. If the weight remains constant across elements of the same basis, then such designs can be interpreted as generalizations of complete sets of mutually unbiased bases, being equivalent whenever the design is composed of d+1 bases in dimension d. We show that, for the purpose of quantum state determination, these designs specify an optimal collection of orthogonal measurements. Using highly nonlinear functions on abelian groups, we construct explicit examples from d+2 orthonormal bases whenever d+1 is a prime power, covering dimensions d=6, 10, and 12, for example, where no complete sets of mutually unbiased bases have thus far been found.Comment: 28 pages, to appear in J. Math. Phy