量子多方同时通信的极限
On the Limits of Quantum Multiparty Simultaneous Communication
- Universidad Adolfo Ibañez（阿道夫·伊巴涅斯大学）
- Universidad de Chile（智利大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本研究通过k方推广的Index Coordination问题，证明量子多方同时通信中公共随机性优势的指数分离，并给出紧的量子下界。
中文摘要 AI 辅助
同时消息传递（SMP）模型为比较经典通信与量子通信提供了基本框架。对于两名玩家，Gavinsky等人（STOC 2006）建立了分离结果，该结果支撑了共享随机性与量子通信的不可比性：\textsc{Index Coordination}问题需要$O(\nlog n)$比特公共硬币，但需要$\Omega(n^{1/3})$量子比特的有界误差。在本工作中，我们通过$\operatorname{IC}_{k,n}$（\textsc{Index Coordination}的自然$k$方推广）建立了多方指数分离。公共硬币协议以最大消息长度$O(\log n)$比特无歧义地解决该问题。相反，无共享纠缠或公共硬币的量子SMP协议在无歧义情形下需要最大消息长度$\Omega(n^{1-1/k})$量子比特，在有界误差情形下需要$\Omega(n^{(k-1)/(k+1)})$量子比特。经典私有硬币协议匹配无歧义界，因此在该情形下量子通信相对于私有随机性不提供渐近优势。对于固定误差参数，所有常数均独立于$k$，从而对每个整数值函数$k=k(n)\ge2$建立指数分离，而不限制其增长。当$k\ge c\log n$（对任意固定$c>0$）时，两个量子下界均变为$\Omega(n)$，匹配全输入协议，并在两种情形下产生紧的线性复杂度。我们的结果表明，量子叠加无法有效模拟公共随机性所提供的协调，将该分离扩展到任意$k$。为了界定多方乘积态的成功概率，我们证明了无歧义量子态识别的精确因子分解定理，该定理可能具有独立的数学意义。
英文摘要
The Simultaneous Message Passing (SMP) model provides a fundamental framework for comparing classical and quantum communication. For two players, Gavinsky et al. (STOC 2006) established a separation underlying the incomparability of shared randomness and quantum communication: \textsc{Index Coordination} needs $O(\log n)$ public-coin bits but $Ω(n^{1/3})$ bounded-error qubits. In this work, we establish a multiparty exponential separation through $\operatorname{IC}_{k,n}$, a natural $k$-party generalization of \textsc{Index Coordination}. Public-coin protocols solve it unambiguously with maximum message length $O(\log n)$ bits. In contrast, quantum SMP protocols without shared entanglement or public coins require maximum message length $Ω(n^{1-1/k})$ qubits in the unambiguous regime and $Ω(n^{(k-1)/(k+1)})$ qubits in the bounded-error regime. A classical private-coin protocol matches the unambiguous bound, so quantum communication provides no asymptotic advantage over private randomness in this regime. For fixed error parameters, all constants are independent of $k$, establishing the exponential separation for every integer-valued function $k=k(n)\ge2$, without restricting its growth. Both quantum lower bounds become $Ω(n)$ when $k\ge c\log n$ for any fixed $c>0$, matching the full-input protocol and yielding tight linear complexity in both regimes. Our results demonstrate that quantum superposition cannot efficiently simulate the coordination afforded by public randomness, extending this separation to arbitrary $k$. To bound success probabilities for multiparty product states, we prove an exact factorization theorem for unambiguous quantum state identification, which may be of independent mathematical interest.
