Non interactive simulation of correlated distributions is decidable

A basic problem in information theory is the following: Let $\mathbf{P} = (\mathbf{X}, \mathbf{Y})$ be an arbitrary distribution where the marginals $\mathbf{X}$ and $\mathbf{Y}$ are (potentially) correlated. Let Alice and Bob be two players where Alice gets samples $\{x_i\}_{i \ge 1}$ and Bob gets samples $\{y_i\}_{i \ge 1}$ and for all $i$, $(x_i, y_i) \sim \mathbf{P}$. What joint distributions $\mathbf{Q}$ can be simulated by Alice and Bob without any interaction? Classical works in information theory by G{\'a}cs-K{\"o}rner and Wyner answer this question when at least one of $\mathbf{P}$ or $\mathbf{Q}$ is the distribution on $\{0,1\} \times \{0,1\}$ where each marginal is unbiased and identical. However, other than this special case, the answer to this question is understood in very few cases. Recently, Ghazi, Kamath and Sudan showed that this problem is decidable for $\mathbf{Q}$ supported on $\{0,1\} \times \{0,1\}$. We extend their result to $\mathbf{Q}$ supported on any finite alphabet. We rely on recent results in Gaussian geometry (by the authors) as well as a new \emph{smoothing argument} inspired by the method of \emph{boosting} from learning theory and potential function arguments from complexity theory and additive combinatorics.Comment: The reduction for non-interactive simulation for general source distribution to the Gaussian case was incorrect in the previous version. It has been rectified no

A doubly exponential upper bound on noisy EPR states for binary games

This paper initiates the study of a class of entangled games, mono-state games, denoted by $(G,\psi)$, where $G$ is a two-player one-round game and $\psi$ is a bipartite state independent of the game $G$. In the mono-state game $(G,\psi)$, the players are only allowed to share arbitrary copies of $\psi$. This paper provides a doubly exponential upper bound on the copies of $\psi$ for the players to approximate the value of the game to an arbitrarily small constant precision for any mono-state binary game $(G,\psi)$, if $\psi$ is a noisy EPR state, which is a two-qubit state with completely mixed states as marginals and maximal correlation less than $1$. In particular, it includes $(1-\epsilon)|\Psi\rangle\langle\Psi|+\epsilon\frac{I_2}{2}\otimes\frac{I_2}{2}$, an EPR state with an arbitrary depolarizing noise $\epsilon>0$.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.Comment: The proof of Lemma C.9 is corrected. The presentation is improved. Some typos are correcte

Decidability of Secure Non-interactive Simulation of Doubly Symmetric Binary Source

Noise, which cannot be eliminated or controlled by parties, is an incredible facilitator of cryptography. For example, highly efficient secure computation protocols based on independent samples from the doubly symmetric binary source (BSS) are known. A modular technique of extending these protocols to diverse forms of other noise without any loss of round and communication complexity is the following strategy. Parties, beginning with multiple samples from an arbitrary noise source, non-interactively, albeit securely, simulate the BSS samples. After that, they can use custom-designed efficient multi-party solutions using these BSS samples. Khorasgani, Maji, and Nguyen (EPRINT--2020) introduce the notion of secure non-interactive simulation (SNIS) as a natural cryptographic extension of concepts like non-interactive simulation and non-interactive correlation distillation in theoretical computer science and information theory. In SNIS, the parties apply local reduction functions to their samples to produce samples of another distribution. This work studies the decidability problem of whether samples from the noise $(X,Y)$ can securely and non-interactively simulate BSS samples. As is standard in analyzing non-interactive simulations, our work relies on Fourier-analytic techniques to approach this decidability problem. Our work begins by algebraizing the simulation-based security definition of SNIS. Using this algebraized definition of security, we analyze the properties of the Fourier spectrum of the reduction functions. Given $(X,Y)$ and BSS with noise parameter $\epsilon$, the objective is to distinguish between the following two cases. (A) Does there exist a SNIS from $BSS(\epsilon)$ to $(X,Y)$ with $\delta$-insecurity? (B) Do all SNIS from $BSS(\epsilon)$ to $(X,Y)$ incur $\delta\u27$-insecurity, where $\delta\u27>\delta$? We prove that there is a bounded computable time algorithm achieving this objective for the following cases. (1) $\delta=O{1/n}$ and $\delta\u27=$ positive constant, and (2) $\delta=$ positive constant, and $\delta\u27=$ another (larger) positive constant. We also prove that $\delta=0$ is achievable only when $(X,Y)$ is another BSS, where $(X,Y)$ is an arbitrary distribution over $\{-1,1\}\times\{-1,1\}$. Furthermore, given $(X,Y)$, we provide a sufficient test determining if simulating BSS samples incurs a constant-insecurity, irrespective of the number of samples of $(X,Y)$. Handling the security of the reductions in Fourier analysis presents unique challenges because the interaction of these analytical techniques with security is unexplored. Our technical approach diverges significantly from existing approaches to the decidability problem of (insecure) non-interactive reductions to develop analysis pathways that preserve security. Consequently, our work shows a new concentration of the Fourier spectrum of secure reduction functions, unlike their insecure counterparts. We show that nearly the entire weight of secure reduction functions\u27 spectrum is concentrated on the lower-degree components. The authors believe that examining existing analytical techniques through the facet of security and developing new analysis methodologies respecting security is of independent and broader interest

Decidability of fully quantum nonlocal games with noisy maximally entangled states

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 $\mathrm{MIP}^*=\mathrm{RE}$ 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 and generalizes the analogous result for nonlocal games. 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.Comment: Reference is update

Dimension Reduction for Polynomials over Gaussian Space and Applications

We introduce a new technique for reducing the dimension of the ambient space of low-degree polynomials in the Gaussian space while preserving their relative correlation structure. As an application, we obtain an explicit upper bound on the dimension of an epsilon-optimal noise-stable Gaussian partition. In fact, we address the more general problem of upper bounding the number of samples needed to epsilon-approximate any joint distribution that can be non-interactively simulated from a correlated Gaussian source. Our results significantly improve (from Ackermann-like to "merely" exponential) the upper bounds recently proved on the above problems by De, Mossel & Neeman [CCC 2017, SODA 2018 resp.] and imply decidability of the larger alphabet case of the gap non-interactive simulation problem posed by Ghazi, Kamath & Sudan [FOCS 2016]. Our technique of dimension reduction for low-degree polynomials is simple and can be seen as a generalization of the Johnson-Lindenstrauss lemma and could be of independent interest

Adaptive microfoundations for emergent macroeconomics

In this paper we present the basics of a research program aimed at providing microfoundations to macroeconomic theory on the basis of computational agentbased adaptive descriptions of individual behavior. To exemplify our proposal, a simple prototype model of decentralized multi-market transactions is offered. We show that a very simple agent-based computational laboratory can challenge more structured dynamic stochastic general equilibrium models in mimicking comovements over the business cycle.Microfoundations of macroeconomics, Agent-based economics, Adaptive behavior