October 7, 2011
Location: Achter de Dom 22 (Utrecht), room 001
Entanglement of completely symmetric quantum states
In quantum information theory, states on a pair of identical quantum systems, which are invariant under global unitary rotations of their (finite dimensional) Hilbert spaces, are known as Werner states. They play a fundamental role in the study of entanglement of pairs of systems. We generalize this notion to n-tuples of identical quantum states, and we find ourselves in the world of classical mathematics, where Schur, Weyl, and Hardy paved the way. Some of the questions concerning entanglement are answered directly in the older literature, some can be answered by the use of it, and some still remain elusive.
A copolymer near a selective interface: variational characterization of the free energy and the critical curve
Consider a polymer chain consisting of a random concatenation of two types of monomers, hydrophobic (favors oil) and hydrophilic (favors water), located near a linear interface separating oil and water. The polymer has an interaction Hamiltonian that rewards matches and penalizes mismatches between monomers and solvents.
The free energy of this model is known to exhibit a crossover between a localized phase (polymer stays close to the interface) and a delocalized phase (polymer moves away from the interface) at a critical curve in the parameter space of interaction strength and disorder bias.
While there are explicit formulas for the free energy and the critical curve in the annealed version of the model, such formulas are missing in the quenched version of the model. The goal of this talk is to report on recent progress towards deriving variational formulas for the latter.
Based on joint work with Erwin Bolthausen and Frank den Hollander.