An orthonormal basis consisting of unentangled (pure tensor) elements in a tensor product of Hilbert spaces is an Unentangled Orthogonal Basis (UOB). In general, for $n$ qubits, we prove that in its natural structure as a real variety, the space of U
OB is a bouquet of products of Riemann spheres parametrized by a class of edge colorings of hypercubes. Its irreducible components of maximum dimension are products of $2^n-1$ two-spheres. Using a theorem of Walgate and Hardy, we observe that the UOB whose elements are distinguishable by local operations and classical communication (called locally distinguishable or LOCC distinguishable UOB) are exactly those in the maximum dimensional components. Bennett et al, in their in-depth study of quantum nonlocality without entanglement, include a specific 3 qubit example UOB which is not LOCC distinguishable; we construct certain generalized counterparts of this UOB in $n$ qubits.
We introduce several homotopy equivalence relations for proper holomorphic mappings between balls. We provide examples showing that the degree of a rational proper mapping between balls (in positive codimension) is not a homotopy invariant. In domain
dimension at least 2, we prove that the set of homotopy classes of rational proper mappings from a ball to a higher dimensional ball is finite. By contrast, when the target dimension is at least twice the domain dimension, it is well known that there are uncountably many spherical equivalence classes. We generalize this result by proving that an arbitrary homotopy of rational maps whose endpoints are spherically inequivalent must contain uncountably many spherically inequivalent maps. We introduce Whitney sequences, a precise analogue (in higher dimensions) of the notion of finite Blaschke product (in one dimension). We show that terms in a Whitney sequence are homotopic to monomial mappings, and we establish an additional result about the target dimensions of such homotopies.
We make several new contributions to the study of proper holomorphic mappings between balls. Our results include a degree estimate for rational proper maps, a new gap phenomenon for convex families of arbitrary proper maps, and an interesting result about inverse images.
