Exploiting Finite Geometries for Better Quantum Advantages in Mermin-Like Games

TL;DR

A new game called the Eloily game is introduced with a quantum–classical success gap of 0.26―, larger than that of the Mermin-Peres and doily games.

Abstract

Quantum games embody non-intuitive consequences of quantum phenomena, such as entanglement and contextuality. The Mermin-Peres game is a simple example, demonstrating how two players can utilise shared quantum information to win a no - communication game with certainty, where classical players cannot. In this paper we look at the geometric structure behind such classical strategies, and borrow ideas from the geometry of symplectic polar spaces to maximise this quantum advantage. We introduce a new game called the Eloily game with a quantum-classical success gap of $0.2\overline{6}$, larger than that of the Mermin-Peres and doily games. We simulate this game in the IBM Quantum Experience and obtain a success rate of $1$, beating the classical bound of $0.7\overline{3}$ demonstrating the efficiency of the quantum strategy.

Figures (8)

• Figure 1: An example of Mermin-Peres Magic Square. Lines denote contexts, which are composed of mutually-commuting operators. Vertical contexts multiply to give $-I_{4}$, while horizontal contexts multiply to give $+I_{4}$. The eigenvalues of the operators multiply to give the eigenvalues of the product - for instance the eigenvalues of the top horizontal context must multiply to $+1$.
• Figure 2: The Mermin-Peres game. Charlie gives Alice and Bob a number $x,y$, who in response return answers $\vec{a} = (a_{1},a_{2},a_{3})$, $\vec{b} = (b_{1},b_{2},b_{3})$. The $a_{i}$ and $b_{i}$ are all selected from the set $\{+1, -1\}$ and conditions on the answers are that $a_{1}a_{2}a_{3} = +1$, $b_{1}b_{2}b_{3} = -1$. The players win iff $a_{y} = b_{x}$.
• Figure 3: An example of a classical strategy in a $3\times 3$ grid: The triplet sent back by Alice corresponds to the row for a given $x\in\{1,2,3\}$ while the triplet sent back by Bob corresponds to the column $y\in\{1,2,3\}$. The bottom-right cell cannot be populated while keeping both row and column sign conditions met. With this strategy they win the game in $8$ cases out of $9$.
• Figure 4: The doily: A $15_3$ point-line configuration that encodes the commutation relations of the two-qubit Pauli group. The red lines are such that the product of the observable is $-I_{4}$ while it is $+I_{4}$ for the other lines. It has degree of contextuality 3.
• Figure 5: A visual representation of the eloily $E_{q}$, a.k.a. $GQ(2,4)$. It is formed by 15 points of a doily $D_{q}$, with a "double-six" 12 points from $E_{q}\backslash D_{q}$. The double-six intersect the doily with representative lines as indicated in the two images. The particular qubit labelling comes from the canonical choice of $q=YYY$. For original image see Polster 1998 Polster98. See also LSVPblunck_GQ_24_veldkamp.

Theorems & Definitions (9)

• Definition 1.1
• Lemma 1.2
• proof
• Proposition 2.1
• proof
• Theorem 3.1
• proof
• Definition 5.1
• Example 5.2