  We present a theory for the single-particle dynamics in binary mixtures with
particle size swaps. The general structure of the theory follows that of the
theory for the collective dynamics in binary mixtures with particle size swaps,
which we developed previously [G. Szamel, Phys. Rev. E 98, 050601(R) (2018)].
Particle size swaps open up an additional relaxation channel, which speeds up
both the collective dynamics and the single-particle dynamics. To make explicit
predictions, we resort to a factorization approximation similar to that
employed in the mode-coupling theory of glassy dynamics. We show that, like in
the standard mode-coupling theory, the single-particle motion becomes arrested
at the dynamic glass transition predicted by the theory for the collective
dynamics. We compare the non-ergodicity parameters predicted by our
mode-coupling-like approach for the an equimolar binary hard sphere mixture
with particle size swaps with the non-ergodicity parameters predicted by the
standard mode-coupling theory for the same system without swaps. Our theory
predicts that the "cage size" is bigger in the system with particle size swaps.
