Quantum resources and protocols are known to outperform their classical counterparts in a variety of communication and information processing tasks. Random access codes (RACs) are one such cryptographically significant family of bipartite communication tasks, wherein the sender encodes a data set (typically a string of input bits) onto a physical system of bounded dimension and transmits it to the receiver, who then attempts to guess a randomly chosen part of the sender's data set (typically one of the sender's input bits). In this work, we introduce a generalization of this task wherein the receiver, in addition to the individual input bits, aims to retrieve randomly chosen functions of the sender's input string. Specifically, we employ sets of mutually unbiased balanced functions, such that perfect knowledge of any one of the constituent functions yields no knowledge about the others. We investigate and bound the performance of (i) classical, (ii) quantum prepare and measure, and (iii) entanglement assisted classical communication (EACC) protocols for the resultant generalized RACs (GRACs). Finally, we detail the case of GRACs with three input bits and find maximal success probabilities for classical, quantum, and EACC protocols, along with the effect of noisy quantum channels on the performance of quantum protocols. Moreover, with the help of this case study, we reveal several characteristic properties of the GRACs which deviate from the standard RACs.

中文翻译：

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相互无偏平衡函数和广义随机访问码

众所周知，量子资源和协议在各种通信和信息处理任务中优于其经典对应物。随机访问码 (RAC) 是一种具有密码意义的双向通信任务家族，其中发送方将数据集（通常是输入位串）编码到一个有界维度的物理系统上，然后将其传输给接收方，然后接收方尝试猜测发送者数据集的随机选择部分（通常是发送者的输入位之一）。在这项工作中，我们介绍了这项任务的泛化，其中接收者除了单个输入位之外，还旨在检索发送者输入字符串的随机选择函数。具体来说，我们采用了一组相互无偏的平衡函数，使得对任何一个组成函数的完美知识不会产生关于其他函数的知识。我们调查并限制了（i）经典，（ii）量子准备和测量，以及（iii）纠缠辅助经典通信（EACC）协议的性能，用于生成的广义 RAC（GRAC）。最后，我们详细介绍了具有三个输入位的 GRAC 的情况，并找到了经典、量子和 EACC 协议的最大成功概率，以及嘈杂的量子信道对量子协议性能的影响。此外，在本案例研究的帮助下，我们揭示了 GRAC 与标准 RAC 不同的几个特性。