We study the one-clean-qubit model of quantum communication where one qubit is in a pure state and all other qubits are maximally mixed. We demonstrate a partial function that has a quantum protocol of cost $O(\log N)$ in this model, however, every interactive randomized protocol has cost $\Omega(\sqrt{N})$, settling a conjecture of Klauck and Lim. In contrast, all prior quantum versus classical communication separations required at least $\Omega(\log N)$ clean qubits. The function demonstrating our separation also has an efficient protocol in the quantum-simultaneous-with-entanglement model of cost $O(\log N )$. We thus recover the state-of-the-art separations between quantum and classical communication complexity. Our proof is based on a recent hypercontractivity inequality introduced by Ellis, Kindler, Lifshitz, and Minzer, in conjunction with tools from the representation theory of compact Lie groups.

翻译：我们研究量子通信中的单干净量子比特模型，其中只有一个量子比特处于纯态，其余所有量子比特均为最大混合态。我们证明存在一个部分函数在该模型下具有代价为$O(\log N)$的量子协议，而任何交互式随机化协议的代价均为$\Omega(\sqrt{N})$，这解决了Klauck和Lim提出的猜想。相比之下，以往所有量子与经典通信分离至少需要$\Omega(\log N)$个干净量子比特。用于证明该分离的函数在量子同步-纠缠模型中也具有代价为$O(\log N)$的高效协议。因此，我们重现了量子与经典通信复杂性之间最前沿的分离结果。我们的证明基于Ellis、Kindler、Lifshitz和Minzer近期引入的某种超压缩性不等式，并结合了紧致李群表示论的相关工具。