AI 中文总结
本文构造了一个有限表示的初等可解群，其 sofic 轮廓接近指数增长，并由此设计了一个量子信道，该信道使用一次成本低但多次使用需近线性内存，揭示了群论与量子信息之间的深刻联系。
AI 中文摘要
一个可解群能离有限群有多远？可解群的每个有限片段都可以用有限集合上的置换来模仿，Cornulier 的 sofic 轮廓统计了在精度 $1/r$ 下这种模仿所需的点数。我们构造了一个有限表示的初等可解群，其有限片段的轮廓为 $\exp(r^{1+o(1)})$，接近我们的计数方法所能检测到的极限。该构造是一个点灯者（lamplighter），其灯放置在构型上而非点上。Houghton 群移动三条射线上的点，有限支撑的仿射映射作用于它们的二元构型，每个构型上有一个 $S_3$ 的副本。于是 $N$ 个点携带 $2^N$ 盏灯，但任意两盏灯可以通过每次至多涉及三个点的移动被带到一起。该群嵌入 Brin 群 $3V$，因此 $3V$ 也有一个近指数轮廓的有限片段。该群还给出了一个显式的量子信道，作用于 $\mathbb C^{873}$，该信道使用一次很便宜，但多次使用很昂贵。一个应用它 $n$ 次的设备，在每次输出释放后才接收下一个输入，当它可花费 $B$ 比特纯度时，需要约 $n/B$ 量子比特的内存，而精确设备在多项式对数因子内达到此界。以最优速率交换时，最小内存约为 $\sqrt n$，同时需要同阶的纯度。该信道是可分解的，并位于具有有限最大混合浴的信道的闭包中，但任何模仿它达到精度 $u$ 的此类浴需要维度 $\exp(u^{-1+o(1)})$，尽管其最小 Stinespring 扩张用一个维度为 $130$ 的纯环境精确实现它。核心是一个一轮定理：在一次使用后，任何此类设备的内存携带该群的近似表示。
英文摘要
How far from finite can an amenable group be? Every finite piece of an amenable group can be imitated by permutations of a finite set, and Cornulier's sofic profile counts how many points such an imitation needs at accuracy $1/r$. We build a finitely presented elementary amenable group with a finite piece whose profile is $\exp(r^{1+o(1)})$, close to the most our counting method can ever detect. The construction is a lamplighter with its lamps on configurations rather than on points. Houghton's group moves the points of three rays, finitely supported affine maps act on their binary configurations, and a copy of $S_3$ sits on every configuration. Then $N$ points carry $2^N$ lamps, yet any two lamps can be brought together by moves that each involve at most three points. The group embeds in Brin's group $3V$, so $3V$ too has a finite piece of nearly exponential profile. The group also gives an explicit quantum channel on $\mathbb C^{873}$ that is cheap to use once and expensive to use many times. A device that applies it $n$ times, releasing each output before the next input arrives, needs about $n/B$ qubits of memory when it may spend $B$ bits of purity, and exact devices achieve this up to polylogarithmic factors. With exchange at the optimal rate, the least memory is about $\sqrt n$, attained with purity of the same order. The channel is factorizable and lies in the closure of channels with finite maximally mixed baths, yet any such bath that imitates it to accuracy $u$ needs dimension $\exp(u^{-1+o(1)})$, although its minimal Stinespring dilation implements it exactly with a pure environment of dimension $130$. The engine is a one-round theorem: after a single use, the memory of any such device carries an approximate representation of the group.
Comments75 pages, 4 figures. The relation lists and verification scripts are included as ancillary files; source and scripts are also archived at https://doi.org/10.5281/zenodo.23050302
