We study the advantages of quantum communication models over classical communication models that are equipped with a limited number of qubits of entanglement. In this direction, we give explicit partial functions on $n$ bits for which reducing the entanglement increases the classical communication complexity exponentially. Our separations are as follows. For every $k\ge 1$: $Q\|^*$ versus $R2^*$: We show that quantum simultaneous protocols with $\tilde{\Theta}(k^5 \log^3 n)$ qubits of entanglement can exponentially outperform two-way randomized protocols with $O(k)$ qubits of entanglement. This resolves an open problem from [Gav08] and improves the state-of-the-art separations between quantum simultaneous protocols with entanglement and two-way randomized protocols without entanglement [Gav19, GRT22]. $R\|^*$ versus $Q\|^*$: We show that classical simultaneous protocols with $\tilde{\Theta}(k \log n)$ qubits of entanglement can exponentially outperform quantum simultaneous protocols with $O(k)$ qubits of entanglement, resolving an open question from [GKRW06, Gav19]. The best result prior to our work was a relational separation against protocols without entanglement [GKRW06]. $R\|^*$ versus $R1^*$: We show that classical simultaneous protocols with $\tilde{\Theta}(k\log n)$ qubits of entanglement can exponentially outperform randomized one-way protocols with $O(k)$ qubits of entanglement. Prior to our work, only a relational separation was known [Gav08].

翻译：我们研究量子通信模型相对于配备有限数量纠缠量子比特的经典通信模型的优势。为此，我们给出了显式的部分函数（定义在$n$比特上），其中减少纠缠会导致经典通信复杂度呈指数级增长。我们的分离结果如下：对于每个$k\ge 1$：$Q\|^*$与$R2^*$对比：我们证明，使用$\tilde{\Theta}(k^5 \log^3 n)$量子比特纠缠的量子同步协议可以指数级优于使用$O(k)$量子比特纠缠的双向随机化协议。这解决了[Gav08]中的一个开放问题，并改进了当前关于有纠缠量子同步协议与无纠缠双向随机化协议之间分离的最新结果[Gav19, GRT22]。$R\|^*$与$Q\|^*$对比：我们证明，使用$\tilde{\Theta}(k \log n)$量子比特纠缠的经典同步协议可以指数级优于使用$O(k)$量子比特纠缠的量子同步协议，解决了[GKRW06, Gav19]中的一个开放问题。我们之前的最佳结果是针对无纠缠协议的关联分离[GKRW06]。$R\|^*$与$R1^*$对比：我们证明，使用$\tilde{\Theta}(k\log n)$量子比特纠缠的经典同步协议可以指数级优于使用$O(k)$量子比特纠缠的随机化单向协议。此前，仅已知一个关联分离结果[Gav08]。