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arXiv:2108.09140 · SIAM Journal on Computing

Nonlocal Games with Noisy Maximally Entangled States are Decidable

Minglong Qin, Penghui Yao

Published Jan 1, 2021• 11 citations
arXivHTMLPDF

Abstract

This paper considers a special class of nonlocal games $(G,\psi)$, where $G$ is a two-player one-round game, and $\psi$ is a bipartite state independent of $G$. In the game $(G,\psi)$, the players are allowed to share arbitrarily many copies of $\psi$. The value of the game $(G,\psi)$, denoted by $\omega^*(G,\psi)$, is the supremum of the winning probability that the players can achieve with arbitrarily many copies of preshared states $\psi$. For a noisy maximally entangled state $\psi$, a two-player one-round game $G$ and an arbitrarily small precision $\epsilon>0$, this paper proves an upper bound on the number of copies of $\psi$ for the players to win the game with a probability $\epsilon$ close to $\omega^*(G,\psi)$. A noisy maximally entangled state is a two-qudit state with both marginals being completely mixed states and the maximal correlation being less than $1$. 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 $\epsilon>0$, where $|\Psi_m\rangle=\frac{1}{\sqrt{m}}\sum_{i=0}^{m-1}|m,m\rangle$ is an $m$-dimensional maximally entangled state. Hence, it is feasible to approximately compute $\omega^*(G,\psi)$ 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 $\mathrm{MIP}^*=\mathrm{RE}$. 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.)

DOI
10.1137/20m134592x

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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−ϵ)∣Ψm⟩⟨Ψm∣+ϵ\mathbbm1mm⊗\mathbbm1mm(1-\epsilon)|\Psi_m\rangle\langle\Psi_m|+\epsilon\frac{\mathbbm{1}_m}{m}\otimes\frac{\mathbbm{1}_m}{m}(1−ϵ)∣Ψm​⟩⟨Ψm​∣+ϵm\mathbbm1m​​⊗m\mathbbm1m​​ 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.)
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.
This paper initiates the study of a class of entangled games, mono-state games, denoted by (G,ψ)(G,ψ)(G,ψ), where GGG is a two-player one-round game and ψψψ is a bipartite state independent of the game GGG. In the mono-state game (G,ψ)(G,ψ)(G,ψ), the players are only allowed to share arbitrary copies of ψψψ. This paper provides a doubly exponential upper bound on the copies of ψψψ for the players to approximate the value of the game to an arbitrarily small constant precision for any mono-state binary game (G,ψ)(G,ψ)(G,ψ), if ψψψ is a noisy EPR state, which is a two-qubit state with completely mixed states as marginals and maximal correlation less than 111. In particular, it includes (1−ε)∣Ψ⟩⟨Ψ∣+εI22⊗I22(1-ε)|Ψ\rangle\langleΨ|+ε\frac{I_2}{2}\otimes\frac{I_2}{2}(1−ε)∣Ψ⟩⟨Ψ∣+ε2I2​​⊗2I2​​, an EPR state with an arbitrary depolarizing noise ε>0ε>0ε>0.The structure of the proofs is built the recent framework about the decidability of the non-interactive simulation of joint distributions, which is completely different from all previous optimization-based approaches or "Tsirelson's problem"-based approaches. This paper develops a series of new techniques about the Fourier analysis on matrix spaces and proves a quantum invariance principle and a hypercontractive inequality of random operators. This novel approach provides a new angle to study the decidability of the complexity class MIP∗^*∗, a longstanding open problem in quantum complexity theory.
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).
We identify a structural mechanism by which perfect play in a nonlocal game requires and self-tests a specific non-maximally entangled state. To this end, we construct a Flagged Magic-Square game by extending the standard Magic-Square game with an additional ``flag'' outcome and two additional questions for each player. The game admits a perfect strategy using a state with Schmidt coefficients proportional to (1,2,2,2,2)(1,\sqrt{2},\sqrt{2},\sqrt{2},\sqrt{2})(1,2​,2​,2​,2​). We give a direct algebraic proof that maximally entangled strategies have winning probability bounded away from one, independently of dimension. Moreover, the game self-tests this state, providing an example of a two-party pseudo-telepathy game that certifies a specific non-maximally entangled state. Our construction illustrates how adding suitable constraints to a rigid measurement geometry can enforce specific Schmidt ratios and provides a modular route to further self-tests based on other rigid pseudo-telepathy games. We establish a general game-to-channel reduction that transfers separations in the game model to one-shot zero-error communication over classical channels. Applied to the Flagged Magic-Square game, it yields a classical channel for which entanglement permits the zero-error transmission of ten messages in a single use, whereas any finite-dimensional maximally entangled resource permits at most nine. Thus, maximally entangled states are not a universally optimal entanglement resource for zero-error communication over classical noisy channels.
We study the impact of noise on the theories of quantum nonlocality and entanglement-assisted communication complexity. We consider non-local games and entanglement-assisted communication complexity in a model where Alice and Bob may share arbitrarily many noisy EPR pairs. We study four noise models: depolarizing noise, unital noise, biased reset noise, and erasure noise. Our results are as follows: 1. We upper bound the value of the CHSH game under all these noise models in terms of the noise parameter, without any assumptions on the measurements used in the strategy. 2. We prove a parallel repetition theorem for general non-local games under all noise models except biased reset noise; we prove an improved parallel repetition theorem for unique games. Our parallel repetition rate is smaller than the quantum parallel repetition rate for CHSH in a nontrivial noise regime. 3. Using our unique-game parallel repetition theorem and a relation defined by the CHSH game, we prove a separation between communication complexity with noisy vs noiseless entanglement. This implies an Ω(n)Ω(n)Ω(n) two-way communication lower bound for distilling nnn EPR pairs from noisy EPR pairs, in the same nontrivial noise regime. 4. We show that any interactive entanglement-assisted communication protocol can be simulated by an SMP communication protocol with noisy shared randomness, with an exponential blowup in communication. This generalizes a known result on the simulation of noiseless shared randomness with noisy shared randomness. 5. We show a polynomial lower bound on the number of copies of noisy EPR pairs required to compute the Equality function with constant communication, under all four noise models. The previous result gives a matching upper bound, and moreover, this answers an open question in the literature on whether logarithmically many noisy shared bits suffice for communication.
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.