抽象代数、射影几何与量子信息的时间编码
Abstract algebra, projective geometry and time encoding of quantum information
摘要：
代数几何概念在量子编码、密码学、层析成像和计算等应用中发挥着越来越重要的作用。我们指出在这里发挥的突出作用伽罗瓦领域视为分圆扩展的整数模的主要特征$p$。它们可用于产生有效的循环编码，用于传输量子密钥，用于量子态恢复和用于量子计算中的纠错。有限射影平面及其推广是分圆概念的几何对应物，它们的坐标化涉及伽罗瓦域，并且它们已被反复用于加密和编码。最后，伽罗瓦域上的性质是生成完备的互无偏基集的基础，这是量子信息处理和量子纠缠的一般概念。伽罗瓦域上的高斯和确保了在这样的协议下的最小不确定性。一些伽罗瓦环是模4整数的分圆扩展，也因为它们在时间编码和相互无偏性中的作用而变得流行。
英文摘要：
Algebraic geometrical concepts are playing an increasing role in quantum applications such as coding, cryptography, tomography and computing. We point out here the prominent role played by Galois fields viewed as cyclotomic extensions of the integers modulo a prime characteristic $p$. They can be used to generate efficient cyclic encoding, for transmitting secrete quantum keys, for quantum state recovery and for error correction in quantum computing. Finite projective planes and their generalization are the geometric counterpart to cyclotomic concepts, their coordinatization involves Galois fields, and they have been used repetitively for enciphering and coding. Finally the characters over Galois fields are fundamental for generating complete sets of mutually unbiased bases, a generic concept of quantum information processing and quantum entanglement. Gauss sums over Galois fields ensure minimum uncertainty under such protocols. Some Galois rings which are cyclotomic extensions of the integers modulo 4 are also becoming fashionable for their role in time encoding and mutual unbiasedness.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
