发表机构
State Key Laboratory of Novel Software Technology, Nanjing University; Hefei National Laboratory（南京大学软件新技术国家重点实验室; 合肥国家实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出信息势这一变分方法，用于分析量子信息复杂度并证明量子通信下界，给出AND函数和集合不相交问题的最优下界及非对称通信复杂度权衡。
AI 中文摘要
量子信息复杂度（QIC）由Touchette [Touchette, STOC 2015]引入，已被证明是证明量子通信复杂度下界的最有力方法之一，并且已被证明等于摊销量子通信复杂度。不幸的是，QIC通常难以分析，因为它是量子条件互信息项之和，而这些项难以估计。在这项工作中，我们引入了一种新的变分方法——信息势，用于分析QIC和证明量子通信下界，其灵感来自量子相对熵的预解表示。该方法将单条消息的QIC与一个二次型联系起来，使其更易于处理，从而使我们能够通过QIC来限制交互式量子协议中势的累积正增量。因此，势增长的下界转化为QIC和量子通信复杂度的下界。作为一个应用，我们给出了两位AND函数的QIC的最优Ω(1/r)下界，以及r轮集合不相交问题的量子通信复杂度的最优Ω(n/r)下界，回答了[Braverman, Garg, Ko, Mao, Touchette FOCS 2015]中的一个开放问题。此外，我们进一步证明了有界轮量子通信复杂度集合不相交问题的直接和定理。利用AND函数QIC的紧下界，我们进一步建立了集合不相交问题非对称量子通信复杂度的近乎紧的权衡：(q_A+1)(q_B+1)=Ω(n)，其中q_A和q_B分别表示Alice和Bob发送的量子比特总数。
英文摘要
Quantum information complexity (QIC), introduced by Touchette [Touchette, STOC 2015], has been shown to be one of the most powerful methods for proving quantum communication complexity and has also been shown to be equal to amortized quantum communication complexity. Unfortunately, QIC is generally hard to analyze because it is a sum of quantum conditional mutual information terms, which are difficult to estimate. In this work, we introduce a new variational approach, the information potential, for analyzing QIC and proving quantum communication lower bounds inspired by the resolvent representation for quantum relative entropy. This approach connects the QIC of individual messages to a quadratic form, making it more amenable and thus enables us to bound the cumulative positive increments of the potential throughout an interactive quantum protocol by its QIC. Lower bounds on the growth of the potential therefore translate into lower bounds on both QIC and quantum communication complexity. As an application, we give an optimal $Ω(1/r)$ lower bound on the QIC of the two-bit $\mathsf{AND}$ function as well as an optimal $Ω(n/r)$ lower bound on the quantum communication complexity of $r$-round Set-Disjointness, answering an open problem in~[Braverman, Garg, Ko, Mao, Touchette FOCS 2015]. Moreover, we further prove a direct-sum theorem for bounded-round quantum communication complexity of Set Disjointness. With the tight bound on the QIC of $\mathsf{AND}$ function, we further establish a nearly tight tradeoff for the asymmetric quantum communication complexity of $\mathsf{Set} \mathsf{Disjointness}$: $(q_A+1)(q_B+1)=Ω(n)$, where $q_A$ and $q_B$ denote the total numbers of qubits sent by Alice and Bob, respectively.
