  Static magnetic field gradients superimposed on the electromagnetic trapping
potential of a Penning trap can be used to implement laser-less spin-motion
couplings that allow the realization of elementary quantum logic operations in
the radio-frequency regime. An important scenario of practical interest is the
application to $g$-factor measurements with single (anti-)protons to test the
fundamental charge, parity, time reversal (CPT) invariance as pursued in the
BASE collaboration [Smorra et al., Eur. Phys. J. Spec. Top. 224, 3055-3108
(2015), Smorra et al., Nature 550, 371-374 (2017), Schneider et al., Science
358, 1081-1084 (2017)]. We discuss the classical and quantum behavior of a
charged particle in a Penning trap with a superimposed magnetic field gradient.
Using analytic and numerical calculations, we find that it is possible to carry
out a SWAP gate between the spin and the motional qubit of a single
(anti-)proton with high fidelity, provided the particle has been initialized in
the motional ground state. We discuss the implications of our findings for the
realization of quantum logic spectroscopy in this system.
