Four-qubit CHSH game

Abstract In this paper, the CHSH quantum game is extended to four players. This is achieved by exploring all possible 4-variable Boolean functions to identify those that yield a game scenario with a quantum advantage using a specific entangled state. Notably, two new four-player quantum games are presented. In one game, the optimal quantum strategy is achieved when players share a $$\vert {GHZ_4} \rangle =\dfrac{1}{\sqrt{2}}(\vert {0000} \rangle +\vert {1111} \rangle )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>G</mml:mi> <mml:mi>H</mml:mi> <mml:msub> <mml:mi>Z</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:mrow> <mml:mrow> <mml:mo>⟩</mml:mo> <mml:mo>=</mml:mo> </mml:mrow> <mml:mstyle> <mml:mfrac> <mml:mn>1</mml:mn> <mml:msqrt> <mml:mn>2</mml:mn> </mml:msqrt> </mml:mfrac> </mml:mstyle> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>|</mml:mo> <mml:mn>0000</mml:mn> <mml:mo>⟩</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:mrow> <mml:mo>|</mml:mo> <mml:mn>1111</mml:mn> <mml:mo>⟩</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> state, breaking the traditional 10% gain observed in 2- and 3-qubit CHSH games and achieving a 22.5% gap. In the other game, players gain a greater advantage using a $$\vert {W_4} \rangle =\dfrac{1}{2}(\vert {0001} \rangle +\vert {0010} \rangle +\vert {0100} \rangle +\vert {1000} \rangle )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> <mml:msub> <mml:mi>W</mml:mi> <mml:mn>4</mml:mn> </mml:msub> <mml:mrow> <mml:mo>⟩</mml:mo> <mml:mo>=</mml:mo> </mml:mrow> <mml:mstyle> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mstyle> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>|</mml:mo> <mml:mn>0001</mml:mn> <mml:mo>⟩</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:mrow> <mml:mo>|</mml:mo> <mml:mn>0010</mml:mn> <mml:mo>⟩</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:mrow> <mml:mo>|</mml:mo> <mml:mn>0100</mml:mn> <mml:mo>⟩</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:mrow> <mml:mo>|</mml:mo> <mml:mn>1000</mml:mn> <mml:mo>⟩</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> state as their quantum resource. Quantum games with other four-qubit entangled states are also explored. To demonstrate the results, these game scenarios are implemented on an online quantum computer, and the advantage of the respective quantum resource for each game is experimentally verified.