有偏随机存取码
Biased Random Access Codes
AI总结：
本文扩展了随机存取码（RAC）协议，允许输入为一般分布，利用数值与分析工具优化有偏RAC性能，提出RAC-tools算法包，并在特定场景下推导上界与自测试条件。
摘要：
随机存取码（RAC）是一种通信任务，其中发送方将随机消息编码为较短的消息，由接收方解码，从而以一定概率恢复原始消息中随机选择的字符。消息和待恢复的字符均假设为均匀分布。在本文中，我们通过允许这些输入具有更一般的分布来扩展此协议，这改变了优化协议性能的编码和解码策略，无论使用经典还是量子资源。我们使用数值和分析工具来解决优化这些有偏RAC性能的问题。在数值方面，我们提出了允许对经典和量子策略的最优性能进行数值评估的算法，并提供了一个旨在实现这些算法的Python包，称为RAC-tools。然后，我们使用此数值工具研究$n^2 \mapsto 1$和$2^d \mapsto 1$场景中单参数族的有偏RAC。对于$n^2 \mapsto 1$场景中的RAC，我们针对输入不相关的情况推导出了一个一般上界，该上界在$n=2$时与量子值一致，且在某些$n=3$的情况下也一致。此外，研究表明达到此上界分别自测试了成对或三个秩为1的投影测量。对于$2^d \mapsto 1$场景中的RAC值，也推导出了类似的上界，并证明如果输入字符串的分布是无偏的，则使用互相无偏测量总是可以达到该上界。
英文摘要：
A random access code (RAC) is a communication task in which the sender encodes a random message into a shorter one to be decoded by the receiver so that a randomly chosen character of the original message is recovered with some probability. Both the message and the character to be recovered are assumed to be uniformly distributed. In this paper, we extend this protocol by allowing more general distributions of these inputs, which alters the encoding and decoding strategies optimizing the protocol performance, with either classical or quantum resources. We approach the problem of optimizing the performance of these biased RACs with both numerical and analytical tools. On the numerical front, we present algorithms that allow a numerical evaluation of the optimal performance over both classical and quantum strategies and provide a Python package designed to implement them, called RAC-tools. We then use this numerical tool to investigate single-parameter families of biased RACs in the $n^2 \mapsto 1$ and $2^d \mapsto 1$ scenarios. For RACs in the $n^2 \mapsto 1$ scenario, we derive a general upper bound for the cases in which the inputs are not correlated, which coincides with the quantum value for $n=2$ and, in some cases for $n=3$. Moreover, it is shown that attaining this upper bound self-tests pairs or triples of rank-1 projective measurements, respectively. An analogous upper bound is derived for the value of RACs in the $2^d \mapsto 1$ scenario, which is shown to be always attainable using mutually unbiased measurements if the distribution of input strings is unbiased.
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