Unitary invariants of qubit systems

We give an algorithm allowing to construct bases of local unitary invariants of pure k-qubit states from the knowledge of polynomial covariants of the group of invertible local filtering operations. The simplest invariants obtained in this way are explicited and compared to various known entanglement measures. Complete sets of generators are obtained for up to four qubits, and the structure of the invariant algebras is discussed in detail.Comment: 19 pages, 1 figur

Derivatives of the identity and generalizations of Milnor's invariants

We synthesize work of U. Koschorke on link maps and work of B. Johnson on the derivatives of the identity functor in homotopy theory. The result can be viewed in two ways: (1) As a generalization of Koschorke's "higher Hopf invariants", which themselves can be viewed as a generalization of Milnor's invariants of link maps in Euclidean space; and (2) As a stable range description, in terms of bordism, of the cross effects of the identity functor in homotopy theory evaluated at spheres. We also show how our generalized Milnor invariants fit into the framework of a multivariable manifold calculus of functors, as developed by the author and Voli\'{c}, which is itself a generalization of the single variable version due to Weiss and Goodwillie.Comment: 25 pages, accepted for publication by the Journal of Topolog

Identifying evolutionary trees and substitution parameters for the general Markov model with invariable sites

The general Markov plus invariable sites (GM+I) model of biological sequence evolution is a two-class model in which an unknown proportion of sites are not allowed to change, while the remainder undergo substitutions according to a Markov process on a tree. For statistical use it is important to know if the model is identifiable; can both the tree topology and the numerical parameters be determined from a joint distribution describing sequences only at the leaves of the tree? We establish that for generic parameters both the tree and all numerical parameter values can be recovered, up to clearly understood issues of `label swapping.' The method of analysis is algebraic, using phylogenetic invariants to study the variety defined by the model. Simple rational formulas, expressed in terms of determinantal ratios, are found for recovering numerical parameters describing the invariable sites