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 demonstrates 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.Computational protocols in which quantum mechanical strategies provide an advantage over classical ones have long been an important focus of study. A way of recasting the Clauser-Horne-Shimony-Holt formulation [1] of Bell's celebrated theorem [2] into a game for which quantum strategies can provide an advantage has been proposed in [3] and has since been referred to as the CHSH game. The players of the game, Alice and Bob, are separated and unable to communicate with each other; each is given one randomly uniform bit, labeled a and b respectively, and they win the game if they return single bits, x and y respectively, such that x⊕y = a·b (mod 2).In game theory, the optimal success probability for a game is called its value, which we denote by ω. The value of the CHSH game, ω(CHSH), depends upon the physics of the systems exploited by Alice and Bob. Famously, if Alice and Bob employ only classical strategies, the value of the CHSH game is ω(CHSH) = 0.75. On the other hand, if they have access to quantum resources, ω(CHSH) = cos 2 ( π 8 ) ≈ 0.85. The limitation on the value of the game for classical systems is called a Bell inequality, and the value 0.75 is often called the Bell bound. The fact that the value of the game when using quantum resources violates the Bell bound, but is nevertheless limited substantially below 1, was first noted by Tsirelson [4], and the value cos 2 ( π 8 ) is known as Tsirelson's bound. Popescu and Rohrlich [5] noted that in more general theories than quantum mechanics, perfect strategies for the CHSH game that achieve a value of 1 could exist via a correlation now known as a Popescu-Rohrlich (PR) box, without violating the no-signaling assumption between Alice and Bob during the game.The CHSH game is of great importance because the dependence of its value from the underlying physical model gives us a tool to distinguish different types of theories experimentally, and allows us to test nature. It also reveals insights into a non-classical feature of quantum mechanics (known colloquially as "non-locality"), which has proven to be a resource for quantum technologies, such as device independent cryptography [6]. Generalisations to...

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