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arXiv:2505.06322

Quantum strategies, error bounds, optimality, and duality gaps for multiplayer XOR, $\mathrm{XOR}^{*}$, compiled XOR, $\mathrm{XOR}^{*}$, and strong parallel repetiton of XOR, $\mathrm{XOR}^{*}$, and FFL games

Pete Rigas

Published May 9, 2025• 0 citations
arXivHTMLPDF

Abstract

We characterize exact, and approximate, optimality of games that players can interact with using quantum strategies. In comparison to a previous work of the author, arXiv: 2311.12887, which applied a 2016 framework due to Ostrev for constructing error bounds beyond CHSH and XOR games, in addition to the existence of well-posed semidefinite programs for determining primal feasible solutions, along with quantum-classical duality gaps, it continues to remain of interest to further develop the construction of error bounds, and related objects, to game-theoretic settings with several participants. In such settings, one encounters a rich information theoretic landscape, not only from the fact that there exists a significantly larger combinatorial space of possible strategies for each player, but also several opportunities for pronounced quantum advantage. We conclude this effort by describing other variants of other possible strategies, as proposed sources for quantum advantage, in $\mathrm{XOR}^{*}$, compiled $\mathrm{XOR}^{*}$, and strong parallel repetition variants of $\mathrm{XOR}^{*}$ games.

DOI
10.48550/arxiv.2505.06322

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Quantum strategies, error bounds, optimality, and duality gaps for multiplayer XOR, XOR∗\mathrm{XOR}^{*}XOR∗, compiled XOR, XOR∗\mathrm{XOR}^{*}XOR∗, and strong parallel repetiton of XOR, XOR∗\mathrm{XOR}^{*}XOR∗, and FFL games
We characterize exact, and approximate, optimality of games that players can interact with using quantum strategies. In comparison to a previous work of the author, arXiv: 2311.12887, which applied a 2016 framework due to Ostrev for constructing error bounds beyond CHSH and XOR games, in addition to the existence of well-posed semidefinite programs for determining primal feasible solutions, along with quantum-classical duality gaps, it continues to remain of interest to further develop the construction of error bounds, and related objects, to game-theoretic settings with several participants. In such settings, one encounters a rich information theoretic landscape, not only from the fact that there exists a significantly larger combinatorial space of possible strategies for each player, but also several opportunities for pronounced quantum advantage. We conclude this effort by describing other variants of other possible strategies, as proposed sources for quantum advantage, in XOR∗\mathrm{XOR}^{*}XOR∗, compiled XOR∗\mathrm{XOR}^{*}XOR∗, and strong parallel repetition variants of XOR∗\mathrm{XOR}^{*}XOR∗ games.
Optimal, and approximately optimal, quantum strategies for XOR∗\mathrm{XOR}^{*}XOR∗ and FFL\mathrm{FFL}FFL games
We analyze optimal, and approximately optimal, quantum strategies in two non-local settings based upon previous investigations of the XOR and CHSH games. By building upon previous arguments due to Ostrev in 2016, {\color{blue}[37]}, which characterized approximately optimal, and optimal, strategies that players Alice and Bob can adopt for increasing their respective winning probabilities, we identify additional applications of the framework for analyzing prospective quantum advantage in other game-theoretic settings. In particular, while it is possible for Alice and Bob to realize quantum advantage if they adopt strategies that are dependent upon entanglement, two-dimensional resource systems, and reversible transformations, the magnitude of the duality gap between the performance of classical and quantum strategies has not yet been classified from the perspective of error bounds. Such bounds are not only dependent upon the winning probability if Alice and Bob leverage entanglement in their strategies but also upon a suitably defined representation-theoretic intertwining operation.
In this work we focus on two classes of games: xor nonlocal games and xor* sequential games with monopartite resources. xor games have been widely studied in the literature of nonlocal games, and we introduce xor* games as their natural counterpart within the class of games where a resource system is subjected to a sequence of controlled operations and a final measurement. Examples of xor* games are $2\ensuremath{\rightarrow}1$ quantum random access codes (QRAC) and the chsh* game introduced by Henaut et al. [Phys. Rev. A 98, 060302 (2018)]. We prove, using the diagrammatic language of process theories, that under certain assumptions these two classes of games can be related via an explicit theorem that connects their optimal strategies, and so their classical (Bell) and quantum (Tsirelson) bounds. We also show that two of such assumptions, the reversibility of transformations and the bidimensionality of the resource system in the xor* games, are strictly necessary for the theorem to hold by providing explicit counterexamples. We conclude with several examples of pairs of xor/xor* games and by discussing in detail the possible resources that power the quantum computational advantages in xor* games.
We obtain quantitative estimates on the decay of the multiplayer optimal value under parallel repetition. In comparison to a previous work of the author in 2025 (arXiv: 2508.09380) which sought to generalize dependency-breaking and anchoring variables from two-player Quantum games, being able to establish quantitative estimates on the decay of the optimal value of a multiplayer game under parallel repetition is of interest to establish under different assumptions. Specifically, independently of the dependency-breaking and anchoring variables that have previously been employed to remove correlations from entangled information shared between Alice and Bob (hence removing dependencies), monotonic concave functions can be used in place of such variables to obtain rates of decay on the optimal value. The game-theoretic setting with two players was first analyzed with monotonic concave functions by Lanzenberger and Maurer. For qi,xi>0q_i , x_i > 0qi​,xi​>0 ∀1≤i≤N\forall 1 \leq i \leq N∀1≤i≤N where $N > 0 $ is the total number of players we adddress an open question raised in their work regarding potential generalizations of two-player monotonic concave functions, through amplification functions of the form ΨMult≡Ψ=N−∏1≤i≤Nexp[−qixi]Ψ_{\textit{Mult}} \equiv Ψ= N - \underset{1 \leq i \leq N}{\prod} \mathrm{exp} \big[ - q_i x_i \big]ΨMult​≡Ψ=N−1≤i≤N∏​exp[−qi​xi​], which in the multiplayer game-theoretic setting have more intricate combinatorial structures.
Nonlocal games are a foundational tool for understanding entanglement and constructing quantum protocols in settings with multiple spatially separated quantum devices. In this work, we continue the study initiated by Kalai et al. (STOC '23) of compiled nonlocal games, played between a classical verifier and a single cryptographically limited quantum device. Our main result is that the compiler proposed by Kalai et al. is sound for any two-player XOR game. A celebrated theorem of Tsirelson shows that for XOR games, the quantum value is exactly given by a semidefinite program, and we obtain our result by showing that the SDP upper bound holds for the compiled game up to a negligible error arising from the compilation. This answers a question raised by Natarajan and Zhang (FOCS '23), who showed soundness for the specific case of the CHSH game. Using our techniques, we obtain several additional results, including (1) tight bounds on the compiled value of parallel-repeated XOR games, (2) operator self-testing statements for any compiled XOR game, and (3) a “nice'' sum-of-squares certificate for any XOR game, from which operator rigidity is manifest.
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 GHZGHZGHZ-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 WWW-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.
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 ∣GHZ4⟩=12(∣0000⟩+∣1111⟩)\vert {GHZ_4} \rangle =\dfrac{1}{\sqrt{2}}(\vert {0000} \rangle +\vert {1111} \rangle )∣GHZ4​⟩=2​1​(∣0000⟩+∣1111⟩) | G H Z 4 ⟩ = 1 2 ( | 0000 ⟩ + | 1111 ⟩ ) 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 ∣W4⟩=12(∣0001⟩+∣0010⟩+∣0100⟩+∣1000⟩)\vert {W_4} \rangle =\dfrac{1}{2}(\vert {0001} \rangle +\vert {0010} \rangle +\vert {0100} \rangle +\vert {1000} \rangle )∣W4​⟩=21​(∣0001⟩+∣0010⟩+∣0100⟩+∣1000⟩) | W 4 ⟩ = 1 2 ( | 0001 ⟩ + | 0010 ⟩ + | 0100 ⟩ + | 1000 ⟩ ) 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.