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An explicit nonlocal game of quantum value one that needs exponential entanglement

Nidhal Mghirbi, Seth Douglas

Published Oct 27, 2026• 1 citation
Publisher

Abstract

How much entanglement does near-optimal play of a two-player nonlocal game require? We give an explicit binary linear system game, with $1687$ equations in $1993$ variables and at most three variables in each equation, whose quantum value is one, and in which every strategy that wins with probability $1-\varepsilon$ needs entanglement entropy $\varepsilon^{-1+o(1)}$, hence local dimension $\exp(\varepsilon^{-1+o(1)})$. The same bound holds for the entanglement of formation of mixed strategies, and a strategy on a maximally entangled state of local dimension $\exp(C\varepsilon^{-1}\log(1/\varepsilon))$ suffices, so the exponent is sharp. For explicit two-player games of quantum value one, the best previous lower bound was polynomial in $1/\varepsilon$. The game comes from a finitely generated, elementary amenable group: a lamplighter whose lamps are Pauli pairs indexed by finite sets of points on the three rays on which Houghton's group acts. Exponentially many of these pairs share one central involution, yet the relation that any two of them commute can be derived from a fixed finite set of relations in a number of steps only quasi-polynomial in the number of points. Two tools may be of independent use. The first counts qubits from approximate Pauli relations measured in an arbitrary state, with the errors of different pairs adding in squares. The second identifies, in both directions and at the natural precision $\sqrt\varepsilon$, the near-optimal strategies of a linear system game with approximate representations of its solution group measured in the players' actual state, so that no power of $\varepsilon$ is lost by passing to a flat state. The same tools give sharp rates for two earlier examples: the CHSH$(n)$ games need $\Theta(\min(n,1/\varepsilon))$ ebits, and the linear system game of Slofstra and Vidick needs local dimension $\Theta(\varepsilon^{-1/2})$.

DOI
10.5281/zenodo.23697254

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The price of precision: usable entanglement in nonlocal games and the geometry of groups

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In a nonlocal game of quantum value one that no finite-dimensional strategy wins, near-perfect play needs growing dimension, which separable states can have. We ask how much usable entanglement it needs: the demand at loss $\varepsilon$ is the least one-way distillable entanglement of the original state of a strategy of loss at most $\varepsilon$. For synchronous games with tested diagonals and for constraint-variable consistency games, demand diverges exactly when no finite-dimensional strategy is perfect; then, for any number of Bell pairs and any fidelity below one, every sufficiently good mixed strategy yields that many at that fidelity by local isometries. For rational synchronous or binary constraint games, a loss guaranteeing this is computable under a promise, but boundedness of demand is undecidable among games of value one. Demand is at least logarithmic if near-winners have a sign relation and an even dilation mirrored by both players, as in a game from Abels' group. For agreement games the directions agree up to a factor three in loss, optimal among Bob-fixed deterministic local symmetrizations. The geometry of the group behind a game, not its presentation, sets the rates. Fixed games demand $\alpha\log(1/\varepsilon)$ for every $\alpha=\frac12\sum_i\log_{m_i}q_i$ with $q_i\ge2$ a power of two dividing $m_i$, a dense set containing all positive rationals and some transcendentals, such as $\frac12\log_62$; a configuration-lamp game demands $\varepsilon^{-1+o(1)}$; Pauli lamps on the points of Houghton's three rays give demand between $\varepsilon^{-0.0690}$ and $\varepsilon^{-1/4}$. If group relations hold exactly and Alice matches Bob to error $e$, demand counts qubits addressable by short words: $\Theta(1/e)$, $\Theta(e^{-k})$ for integers $k\ge1$, or $2^{\Theta(1/e)}$. Conditional on our attainment and tail bounds, the $I_{3322}$ optimum needs bounded usable entanglement.

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How much entanglement does near-optimal play of a two-player nonlocal game require? We give an explicit binary linear system game, with 168716871687 equations in 199319931993 variables and at most three variables in each equation, whose quantum value is one, and in which every strategy that wins with probability 1−ε1-\varepsilon1−ε needs entanglement entropy ε−1+o(1)\varepsilon^{-1+o(1)}ε−1+o(1), hence local dimension exp⁡(ε−1+o(1))\exp(\varepsilon^{-1+o(1)})exp(ε−1+o(1)). The same bound holds for the entanglement of formation of mixed strategies, and a strategy on a maximally entangled state of local dimension exp⁡(Cε−1log⁡(1/ε))\exp(C\varepsilon^{-1}\log(1/\varepsilon))exp(Cε−1log(1/ε)) suffices, so the exponent is sharp. For explicit two-player games of quantum value one, the best previous lower bound was polynomial in 1/ε1/\varepsilon1/ε. The game comes from a finitely generated, elementary amenable group: a lamplighter whose lamps are Pauli pairs indexed by finite sets of points on the three rays on which Houghton's group acts. Exponentially many of these pairs share one central involution, yet the relation that any two of them commute can be derived from a fixed finite set of relations in a number of steps only quasi-polynomial in the number of points. Two tools may be of independent use. The first counts qubits from approximate Pauli relations measured in an arbitrary state, with the errors of different pairs adding in squares. The second identifies, in both directions and at the natural precision ε\sqrt\varepsilonε​, the near-optimal strategies of a linear system game with approximate representations of its solution group measured in the players' actual state, so that no power of ε\varepsilonε is lost by passing to a flat state. The same tools give sharp rates for two earlier examples: the CHSH(n)(n)(n) games need Θ(min⁡(n,1/ε))\Theta(\min(n,1/\varepsilon))Θ(min(n,1/ε)) ebits, and the linear system game of Slofstra and Vidick needs local dimension Θ(ε−1/2)\Theta(\varepsilon^{-1/2})Θ(ε−1/2).
In a nonlocal game of quantum value one that no finite-dimensional strategy wins, near-perfect play needs growing dimension, which separable states can have. We ask how much usable entanglement it needs: the demand at loss ε\varepsilonε is the least one-way distillable entanglement of the original state of a strategy of loss at most ε\varepsilonε. For synchronous games with tested diagonals and for constraint-variable consistency games, demand diverges exactly when no finite-dimensional strategy is perfect; then, for any number of Bell pairs and any fidelity below one, every sufficiently good mixed strategy yields that many at that fidelity by local isometries. For rational synchronous or binary constraint games, a loss guaranteeing this is computable under a promise, but boundedness of demand is undecidable among games of value one. Demand is at least logarithmic if near-winners have a sign relation and an even dilation mirrored by both players, as in a game from Abels' group. For agreement games the directions agree up to a factor three in loss, optimal among Bob-fixed deterministic local symmetrizations. The geometry of the group behind a game, not its presentation, sets the rates. Fixed games demand αlog⁡(1/ε)\alpha\log(1/\varepsilon)αlog(1/ε) for every α=12∑ilog⁡miqi\alpha=\frac12\sum_i\log_{m_i}q_iα=21​∑i​logmi​​qi​ with qi≥2q_i\ge2qi​≥2 a power of two dividing mim_imi​, a dense set containing all positive rationals and some transcendentals, such as 12log⁡62\frac12\log_6221​log6​2; a configuration-lamp game demands ε−1+o(1)\varepsilon^{-1+o(1)}ε−1+o(1); Pauli lamps on the points of Houghton's three rays give demand between ε−0.0690\varepsilon^{-0.0690}ε−0.0690 and ε−1/4\varepsilon^{-1/4}ε−1/4. If group relations hold exactly and Alice matches Bob to error eee, demand counts qubits addressable by short words: Θ(1/e)\Theta(1/e)Θ(1/e), Θ(e−k)\Theta(e^{-k})Θ(e−k) for integers k≥1k\ge1k≥1, or 2Θ(1/e)2^{\Theta(1/e)}2Θ(1/e). Conditional on our attainment and tail bounds, the I3322I_{3322}I3322​ optimum needs bounded usable entanglement.
This paper considers a special class of nonlocal games (G,ψ)(G,\psi)(G,ψ), where GGG is a two-player one-round game, and ψ\psiψ is a bipartite state independent of GGG. In the game (G,ψ)(G,\psi)(G,ψ), the players are allowed to share arbitrarily many copies of ψ\psiψ. The value of the game (G,ψ)(G,\psi)(G,ψ), denoted by ω∗(G,ψ)\omega^*(G,\psi)ω∗(G,ψ), is the supremum of the winning probability that the players can achieve with arbitrarily many copies of preshared states ψ\psiψ. For a noisy maximally entangled state ψ\psiψ, a two-player one-round game GGG and an arbitrarily small precision ϵ>0\epsilon>0ϵ>0, this paper proves an upper bound on the number of copies of ψ\psiψ for the players to win the game with a probability ϵ\epsilonϵ close to ω∗(G,ψ)\omega^*(G,\psi)ω∗(G,ψ). A noisy maximally entangled state is a two-qudit state with both marginals being completely mixed states and the maximal correlation being less than 111. In particular, it includes $(1-\epsilon)|\Psi_m\rangle\langle\Psi_m|+\epsilon\frac{\mathbbm{1}_m}{m}\otimes\frac{\mathbbm{1}_m}{m}$ for ϵ>0\epsilon>0ϵ>0, where ∣Ψm⟩=1m∑i=0m−1∣m,m⟩|\Psi_m\rangle=\frac{1}{\sqrt{m}}\sum_{i=0}^{m-1}|m,m\rangle∣Ψm​⟩=m​1​∑i=0m−1​∣m,m⟩ is an mmm-dimensional maximally entangled state. Hence, it is feasible to approximately compute ω∗(G,ψ)\omega^*(G,\psi)ω∗(G,ψ) to an arbitrary precision. Recently, a breakthrough result by Ji et al. showed that it is undecidable to approximate the values of nonlocal games to a constant precision, when the players preshare arbitrarily many copies of perfect maximally entangled states, which implies that MIP∗=RE\mathrm{MIP}^*=\mathrm{RE}MIP∗=RE. In contrast, our result implies the hardness of approximating nonlocal games collapses when the preshared maximally entangled states are noisy. The paper develops a theory of Fourier analysis on matrix spaces by extending a number of techniques in Boolean analysis and Hermitian analysis to matrix spaces. We establish a series of new techniques, such as a quantum invariance principle and a hypercontractive inequality for random operators, which we believe have further applications. (A corrected version is attached.)
We describe a two-player non-local game, with a fixed small number of questions and answers, such that an ϵ -close to optimal strategy requires an entangled state of dimension 2 Ω ( ϵ − 1 / 8 ) . Our non-local game is inspired by the three-player non-local game of Ji, Leung and Vidick \cite{ji2018three}. It reduces the number of players from three to two, as well as the question and answer set sizes. Moreover, it provides an (arguably) elementary proof of the non-closure of the set of quantum correlations, based on embezzlement and self-testing. In contrast, previous proofs \cite{slofstra2019set, dykema2017non, musat2018non} involved representation theoretic machinery for finitely-presented groups and C ∗ -algebras.
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​.
Can uncomputability occur in the resource law of one fixed finite nonlocal game, rather than in a problem whose input is a game? We construct a finite two-player Boolean constraint-system game with rational question probabilities, quantum value one, and no finite-dimensional perfect strategy. If D(ε)D(\varepsilon)D(ε) is the least local dimension needed to win with probability at least 1−ε1-\varepsilon1−ε, then lim⁡n→∞log⁡2D(2−n)/n=(3+θK)/4\lim_{n\to\infty}\log_2 D(2^{-n})/n=(3+\theta_K)/4limn→∞​log2​D(2−n)/n=(3+θK​)/4, where θK=∑e∈K4−(e+1)\theta_K=\sum_{e\in K}4^{-(e+1)}θK​=∑e∈K​4−(e+1) and KKK is the halting set. This exponent has the Turing degree of KKK. The construction combines finite operator relations for three Cantor clocks, a counter program with forced initialization, and a conditional clock that charges dimension for halting inputs. A Boolean compiler preserves weighted precision linearly and converts flat models to strategies with quadratic loss. The lower bound holds in every density matrix. Explicit finite models attain the same rate. The flag-density lower bound uses compactness without an effective modulus; convergence to the dimension exponent has no computable certified rate.
Abstract This paper considers the decidability of fully quantum nonlocal games with noisy maximally entangled states. Fully quantum nonlocal games are a generalization of nonlocal games, where both questions and answers are quantum and the referee performs a binary POVM measurement to decide whether they win the game after receiving the quantum answers from the players. The quantum value of a fully quantum nonlocal game is the supremum of the probability that they win the game, where the supremum is taken over all the possible entangled states shared between the players and all the valid quantum operations performed by the players. The seminal work MIP∗=RE\mathrm{MIP}^*=\mathrm{RE}MIP∗=RE~\cite{JNVWY'20,JNVWYuen'20} implies that it is undecidable to approximate the quantum value of a fully nonlocal game. This still holds even if the players are only allowed to share (arbitrarily many copies of) maximally entangled states. This paper investigates the case that the shared maximally entangled states are noisy. We prove that there is a computable upper bound on the copies of noisy maximally entangled states for the players to win a fully quantum nonlocal game with a probability arbitrarily close to the quantum value. This implies that it is decidable to approximate the quantum values of these games. Hence, the hardness of approximating the quantum value of a fully quantum nonlocal game is not robust against the noise in the shared states. This paper is built on the framework for the decidability of non-interactive simulations of joint distributions~\cite{7782969,doi:10.1137/1.9781611975031.174,Ghazi:2018:DRP:3235586.3235614} and generalizes the analogous result for nonlocal games in~\cite{qin2021nonlocal}. We extend the theory of Fourier analysis to the space of super-operators and prove several key results including an invariance principle and a dimension reduction for super-operators. These results are interesting in their own right and are believed to have further applications.