  We introduce a simple single-system game inspired by the
Clauser-Horne-Shimony-Holt (CHSH) game. For qubit systems subjected to unitary
gates and projective measurements, we prove that any strategy in our game can
be mapped to a strategy in the CHSH game, which implies that Tsirelson's bound
also holds in our setting. More generally, we show that the optimal success
probability depends on the reversible or irreversible character of the gates,
the quantum or classical nature of the system and the system dimension. We
analyse the bounds obtained in light of Landauer's principle, showing the
entropic costs of the erasure associated with the game. This shows a connection
between the reversibility in fundamental operations embodied by Landauer's
principle and Tsirelson's bound, that arises from the restricted physics of a
unitarily-evolving single-qubit system.
