EPJ Quantum Technol. (2014) 1: 11
https://doi.org/10.1140/epjqt11

Research

A dynamic systems approach to fermions and their relation to spins

¹ Institute for Scientific Interchange Foundation, Via Alassio 11/c, Torino, 10126, Italy
² Department of Theoretical Physics, University of the Basque Country UPV/EHU, P.O. Box 644, Bilbao, E-48080, Spain
³ Department of Computer Science, University College London, Gower St, London, WC1E 6BT, UK
⁴ Department Chemie, Technische Universität München, Lichtenbergstrasse 4, Garching, 85747, Germany
⁵ Zentrum Mathematik, M5, Technische Universität München, Boltzmannstrasse 3, Garching, 85748, Germany

^* e-mail: robert.zeier@ch.tum.de

Received: 8 April 2014
Accepted: 21 July 2014
Published online: 11 September 2014

Abstract

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.