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arXiv:2403.09512

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

Colm Kelleher, Frédéric Holweck, Péter Lévay

Published Mar 14, 2024• 0 citations
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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.

DOI
10.48550/arxiv.2403.09512

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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 0.26‾0.2\overline{6}0.26, 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 111, beating the classical bound of 0.73‾0.7\overline{3}0.73 demonstrating the efficiency of the quantum strategy.
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 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 3 ― demonstrating the efficiency of the quantum strategy.
Abstract Quantum pseudo-telepathy games, such as the Mermin–Peres magic square and the doily game (DG), theoretically allow players to win with unit probability when using entangled quantum strategies. We quantitatively characterize the quantum advantage in these games and compare it with violations of two Bell inequalities: the Clauser–Horne–Shimony–Holt and the Collins–Gisin inequalities. The analysis is restricted to two families of two-qubit states: modified Werner states and Bell-diagonal states. For each case, we identify and quantify the regions of quantum state space that exhibit either a quantum advantage or a Bell inequality violation, relative to the set of all entangled states. Within these families, the DG encompasses a larger fraction of entangled states than the Mermin–Peres magic square game, although both cover significantly smaller regions of entangled states compared to those where Bell inequalities are violated. Although both approaches are fundamentally linked to quantum contextuality, our analysis of the examined two-qubit state families suggests that Bell inequalities are more effective at revealing entanglement, even if pseudo-telepathy games offer a more intuitive and conceptually appealing perspective.
We present a generalization of the Mermin-Peres magic square game to NNN players, in which the square is replaced by an NNN-dimensional hypercube. We show that the winning probability of the optimal classical strategy remains 8/98/98/9 for all NNN, while for N≥3N \ge 3N≥3, a generalization of the quantum strategy for N=2N = 2N=2 achieves a winning probability between 1/31/31/3 and 3/43/43/4. This loss of quantum advantage for N≥3N \ge 3N≥3 is a consequence of monogamy.
It is known that Mermin-Peres like proofs of quantum contextuality can furnish non-local games with a guaranteed quantum strategy, when classically no such guarantee can exist. This phenomenon, also called quantum pseudo-telepathy, has been studied in the case of the so-called Mermin Magic square game. In this paper we review in detail two different ways of implementing on a quantum computer such a game and propose a new Doily game based on the geometry of 2-qubit Pauli group. We show that the quantumness of these games are almost revealed when we play them on the IBM Quantum Experience, however the inherent noise in the available quantum machines prevents a full demonstration of the non-classical aspects.
The Mermin-Peres magic square game is a cooperative two-player nonlocal game in which shared quantum entanglement allows the players to win with certainty, while players limited to classical operations cannot do so, a phenomenon dubbed "quantum pseudo-telepathy". The game has a referee separately ask each player to color a subset of a 3x3 grid. The referee checks that their colorings satisfy certain parity constraints that can't all be simultaneously realized. We define a generalization of these games to be played on an arbitrary arrangement of intersecting sets of elements. We characterize exactly which of these games exhibit quantum pseudo-telepathy, and give quantum winning strategies for those that do. In doing so, we show that it suffices for the players to share three Bell pairs of entanglement even for games on arbitrarily large arrangements. Moreover, it suffices for Alice and Bob to use measurements from the three-qubit Pauli group. The proof technique uses a novel connection of Mermin-style games to graph planarity.
Prior to this work, the largest known gap between the entangled value and the classical value for a one-round two-player nonlocal game with a perfect entangled strategy using two Bell states of entanglement was 19\frac{1}{9}91​, achieved by the Mermin-Peres magic square game. A larger gap of 215\frac{2}{15}152​ has been claimed for the related doily game of Kelleher, Roomy, and Holweck (Quant. Inf. Proc., 2024). However, this value assumes that the classical players use identical local strategies, and we show that dropping this assumption reduces the gap back to 19\frac{1}{9}91​. We then demonstrate that 19\frac{1}{9}91​ is not optimal by explicitly constructing a new nonlocal game with gap 435\frac{4}{35}354​.