Narrow Results

We study quantum state transfer through a qubit network modeled by spins with XY interaction, when relying on a single excitation. We show that it is possible to achieve perfect transfer by shifting (adding) energy to specific vertices. This technique appears to be a potentially powerful tool for changing, and in some cases improving, the transfer capabilities of quantum networks. Analytical results are presented for all-to-all networks and for all-to-all networks with a missing link. Moreover, we evaluate the effect of random fluctuations on the transmission fidelity.

We prove an upper bound for the evaluation of all classical SU₂ spin networks conjectured by Garoufalidis and van der Veen. This implies one half of the analogue of the volume conjecture which they proposed for classical spin networks. We are also able to obtain the other half, namely, an exact determination of the spectral radius, for the special class of generalized drum graphs. Our proof uses a version of Feynman diagram calculus which we developed as a tool for the interpretation of the symbolic method of classical invariant theory, in a manner which is rigorous yet true to the spirit of the classical literature.

In this talk, we elaborate on the operation of graph contraction introduced by Gurau in his study of the Schwinger-Dyson equations. After a brief review of colored tensor models, we identify the Lie algebra appearing in the Schwinger-Dyson equations as a Lie algebra associated to a Hopf algebra of the Connes-Kreimer type. Then, we show how this operation also leads to an analogue of the Wilsonian flow for the effective action. Finally, we sketch how this formalism may be adapted to group field theories.