2. Introduction to Linear Constraint System Games
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A Linear Constraint System (LCS) game is a type of nonlocal game that can be used to demonstrate quantum advantage. In an LCS game, two players, Alice and Bob, are given inputs and must produce outputs that satisfy certain linear constraints. The players are not allowed to communicate with each other during the game, but they can share entangled quantum states beforehand.
Alice generally receives an input that corresponds to a linear constraint, and she must produce an output that satisfies that constraint. Bob receives an input that corresponds to a variable in the constraints, and he must produce an output that is consistent with the constraints given Alice's outpt.
2.1. Mermin Peres Magic Square Game
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The most famous example of an LCS game is the Mermin Peres Magic Square game. In this game, Alice and Bob are given a 3x3 grid of variables, and they must fill in the grid with 0s and 1s such that each row and column sums to an even number. Classically, it is impossible for Alice and Bob to win this game with probability 1, but if they share a specific entangled quantum state, they can win with probability 1.