We introduce a three-player nonlocal game, with a finite number of classical questions and answers, such that the optimal success probability of 1 in the game can only be achieved in the limit of strategies using arbitrarily high-dimensional entangled states. Precisely, there exists a constant 0
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.
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−ε needs entanglement entropy ε−1+o(1), hence local dimension 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)) suffices, so the exponent is sharp. For explicit two-player games of quantum value one, the best previous lower bound was polynomial in 1/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, 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 is lost by passing to a flat state. The same tools give sharp rates for two earlier examples: the CHSH(n) games need Θ(min(n,1/ε)) ebits, and the linear system game of Slofstra and Vidick needs local dimension Θ(ε−1/2).
Several variants of nonlocal games have been considered in the study of quantum entanglement and nonlocality. This paper concerns two of these variants, called quantum-classical games and extended nonlocal games. We give a construction of an extended nonlocal game from any quantum-classical game that allows one to translate certain facts concerning quantum- classical games to extended nonlocal games. In particular, based on work of Regev and Vidick, we conclude that there exist extended nonlocal games for which no finite-dimensional entangled strategy can be optimal. While this conclusion is a direct consequence of recent work of Slofstra, who proved a stronger, analogous result for ordinary (non-extended) nonlocal games, the proof based on our construction is considerably simpler, and the construction itself might potentially have other applications in the study of entanglement and nonlocality.
Experimental tests of shallow-circuit quantum advantage require explicit classical bounds at finite circuit sizes. We refine the finite-size classical soundness bound of Aasnaess's graph-distributed construction to depend linearly on the number of players. Combined with standard disjoint-player repetition, this gives a two-round test on a single processor for any finite nonlocal game with a finite-dimensional perfect quantum strategy and classical winning probability γ<1, with arbitrarily small classical success. The method is based on playing m copies with disjoint players and teleporting each player's register to a uniformly random one of N sites before the questions are revealed. Each answer bit of a depth-D, fan-in-K classical response with fixed wiring depends on at most KD question wires, making cross-player dependencies unlikely when N is large. The quantum implementation has constant depth per round and wins with certainty. A classical device wins with probability at most γm+O(mKD/N), which vanishes as O(logN/N) for m=⌈log1/γN⌉ at fixed D and K. We present an explicit proposal for an experimentally testable quantum advantage with 99 qubits.
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~\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.
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 is the least one-way distillable entanglement of the original state of a strategy of loss at most ε. 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) for every α=21∑ilogmiqi with qi≥2 a power of two dividing mi, a dense set containing all positive rationals and some transcendentals, such as 21log62; a configuration-lamp game demands ε−1+o(1); Pauli lamps on the points of Houghton's three rays give demand between ε−0.0690 and ε−1/4. If group relations hold exactly and Alice matches Bob to error e, demand counts qubits addressable by short words: Θ(1/e), Θ(e−k) for integers k≥1, or 2Θ(1/e). Conditional on our attainment and tail bounds, the I3322 optimum needs bounded usable entanglement.