CIF: Small: Fundamental limits in ambiguous communication

噪声是信息通道不可避免的特征，无论是由环境引起的，还是由于固有的随机性，都是用于在两个或更多方之间传输数据的信息通道。这种信道的所谓零错误容量是指在允许信道多次使用时，以零错误概率通过信道传输信息的最佳速率。虽然在实践中，噪声的影响随时间而变化，但迄今为止可用的容量理论主要集中在简化的情况下，其中信道的连续使用是独立的。该项目旨在建立理论基础，以便在状态依赖状态下估计零误差容量和密切相关的参数，其中通道的每个单独使用取决于其先前的使用和周围噪声的当前状态。它正在使用开发的框架来研究信息渠道的特殊情况，这被视为游戏的策略-称为双证明者游戏-在游戏中，两个玩家通过问答过程接受验证者的测试，以共同拥有某一特定知识。到目前为止，这种游戏只在无记忆的情况下被考虑，即连续的游戏回合彼此独立进行。该项目正在研究两个证明者游戏的胜率，其中玩家和验证者拥有记忆，游戏规则——被测试的知识——可能会随着时间的推移而演变。在信息通道设置中，本项目对有内存通道的零错误容量进行定义和研究，作为特例，将无内存通道通常的零错误容量作为一次性参数。这是通过引入状态相关通道的混淆图来实现的，而该参数的估计和界限是通过开发众所周知的Lovász有限图数的可测量版本来获得的。在两个证明者游戏的设置中，该项目正在开发具有记忆的玩家之间的无信号相关性，包括有用的子类，如量子相关性类，作为具有有限问答集的游戏玩家之间无信号相关性的深远推广。这种方法允许在一般框架中考虑值分离问题和识别参与者的渐近行为，允许应用泛函分析和算子理论工具。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Noise, either arising from the environment or due to inherent randomness, is an unavoidable feature of information channels, which are used to transmit data between two or more parties. So-called zero-error capacities of such channels are the optimal rates at which information can be transmitted via the channel with a zero probability of error when multiple uses of the channel are allowed. While in practice the effects of noise change with time, the capacity theories available until now focus mostly on the simplified picture where the successive uses of the channel are independent. This project aims at establishing the theoretical foundations that would allow the estimation of zero-error capacities and closely related parameters in the state-dependent regime, where each individual use of the channel depends on its previous uses and the current state of the surrounding noise. It is using the developed framework to study a special case of information channels, which are viewed as strategies for games -- called two-prover games -- in which two players are tested by a verifier, through a question-answer process, for the joint possession of a certain piece of knowledge. Such games have until now been considered only in the memoryless situation where the successive rounds of the game are played independently from each other. The project is studying the winning rates of two-prover games, where the players and the verifier possess memory and the rules of the game -- the tested knowledge -- may evolve with time.In the setting of information channels, the project is defining and studying the zero-error capacity of a channel with memory, which covers as a special case the usual zero-error capacity of a memoryless channel as a one-shot parameter. This is being achieved through the introduction of the confusability graph of a state-dependent channel, while estimations and bounds for this parameter are obtained by developing a measurable version of the well-known Lovász number of a finite graph. In the setting of two prover games, the project is developing no-signaling correlations between players with memory, including useful subclasses such as the class of quantum correlations, as a far-reaching generalization of no-signaling correlations between players of games with finite question-answer sets. This approach allows the consideration of value-separation questions and the identification of the players' asymptotic behavior in a general framework, allowing the application of functional analytic and operator theoretic tools.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.