  The separability of the spatial modes of a charged particle in a Penning trap
in the presence of an environment is studied by means of the positive partial
transpose (PPT) criterion. Assuming a weak Markovian environment, described by
linear Lindblad operators, our results strongly suggest that the environmental
coupling of the axial and cyclotron degrees of freedom does not lead to
entanglement at experimentally realistic temperatures. We therefore argue that,
apart from unavoidable decoherence, the presence of such an environment does
not alter the effectiveness of recently suggested quantum information protocols
in Penning traps, which are based on the combination of a spatial mode with the
spin of the particle.
