A dynamic systems approach to fermions and their relation to spins

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.

Erratum: “Symmetry principles in quantum systems theory” [J. Math. Phys. 52, 113510 (2011)]

“…7) which contains for sp(n) the cases sp(i) ⊕ sp(n − i) with n > 1 and i < n as well as u(n) with n > 1. As the dimension of g is strictly larger than the dimensions for these cases (assuming that n > 2), we conclude that the existence of g rules out the possibility of (b) except for n = 2 where the cases (a) and (b) coincide as sp (2) so (5). Similarly, the case (d) is impossible and for n = 1, the cases (a) and (c) are isomorphic.…”

On the partial transpose of fermionic Gaussian states

“…The product of all Majorana operators define the parity operator, = ∏ = P a i N n N n 1 2 , which plays an important role in fermionic systems. According to the parity superselection rule, only density matrices that commute with P correspond to physical states [34][35][36].…”

Spectral properties of reduced fermionic density operators and parity superselection rule

“…One may expect that the transfer of quantum information through a fermionic channel would be different from that for bosons [39], as the fermionic theory differs from the qubit theory [40,41] in both tomography [42] and quantum computation [43].…”