A dynamic systems approach to fermions and their relation to spins

The key dynamic properties of fermionic systems, like controllability, reachability, and simulability, are investigated in a general Lie-theoretical frame for quantum systems theory. It just requires knowing drift and control Hamiltonians of an experimental set-up. Then one can easily determine all the states that can be reached from any given initial state. Likewise all the quantum operations that can be simulated with a given set-up can be identified. Observing the parity superselection rule, we treat the fully controllable and quasifree cases of fermions, as well as various translation-invariant and particle-number conserving cases. We determine the respective dynamic system Lie algebras to express reachable sets of pure (and mixed) states by explicit orbit manifolds. PACS Codes: 03.67.Ac, 02.30.Yy, 75.10.Pq.