Universal quantum control in irreducible state-space sectors: Application to bosonic and spin-boson systems

We analyze the dynamical-algebraic approach to universal quantum control introduced in P. Zanardi and S. Lloyd, e-print quant-ph/0305013. The quantum state space $\mathcal{H}$ encoding information decomposes into irreducible sectors and subsystems associated with the group of available evolutions. If this group coincides with the unitary part of the group algebra $C\mathcal{K}$ of some group $\mathcal{K}$ then universal control is achievable over the K-irreducible components of $\mathcal{H}.$ This general strategy is applied to different kinds of bosonic systems. We first consider massive bosons in a double well and show how to achieve universal control over all finite-dimensional Fock sectors. We then discuss a multimode massless case giving the conditions for generating the whole infinite-dimensional multimode Heisenberg-Weyl enveloping algebra. Finally we show how to use an auxiliary bosonic mode coupled to finite-dimensional systems to generate high-order nonlinearities needed for universal control.