Noncommutative Day 07

Friday 07 December 2007

Department of Mathematics

Imperial College

Room: Huxley Building

658 between 1-4 p.m., and then room 340 from 4 p.m.

Nilanjana Datta (statslab.cam.ac.uk)

TITLE: Perfect transfer of Quantum Information across Spin Networks

ABSTRACT:
Quantum information is encoded in quantum mechanical states of a physical system. Hence, reliable transmission of quantum information from one location to another entails the perfect transfer of quantum mechanical states between these locations. We consider the situation in which the underlying system used for this information transmission is a system of N interacting spins and address the problem of arranging the spins in a network in a manner which would allow perfect state transfer over the largest possible distance. The network is described by a graph G, with the vertices representing the locations of the spins and the edges connecting spins which interact with each other. State transfer is achieved by the time evolution of the spin system under a suitable Hamiltonian. This can be equivalently viewed as a continuous-time quantum walk on the graph G. We find the maximal distance of perfect state transfer and prove that the corresponding quantum walk exhibits an exponential speed-up over its classical counterpart.

Adam Skalski (lancaster.ac.uk)

TITLE: Nonsymmetric Dirichlet forms in the noncommutative context

ABSTRACT: Beginning from the work of Beurling and Deny in the 1950s classical quadratic Dirichlet forms were used to characterise and analyse symmetric Markov semigroups via their generators. In recent years many results of the symmetric theory have been extended to Markov semigroups whose adjoint semigroups are also Markov (see the monograph of Ma and Rockner, 1993) . Here we discuss non-symmetric Dirichlet forms in the fully noncommutative context, where a von Neumann algebra M equipped with a faithful normal state replaces a classical L-infinity space with a reference measure. Beurling-Deny type conditions are formulated to characterise these sesquilinear forms on L2(M) which lead to Markov semigroups on M whose KMS-adjoints are also Markov. Some examples related to inner derivations are also provided.

Chris Barnett (imperial.ac.uk)

TITLE: Random Stopping and Martingale Representation.

ABSTRACT: