🤖 AI Summary
This work systematically characterizes how bounded entanglement resources govern the trade-off between quantum and classical communication complexity. For three fundamental model pairs—quantum simultaneous protocols versus two-way randomized protocols, classical simultaneous protocols versus quantum simultaneous protocols, and classical simultaneous protocols versus randomized one-way protocols—we construct explicit *n*-bit partial functions that achieve exponential separations in communication complexity under tight entanglement constraints: *O*(1), *O*(log *n*), and *O*(*n*) qubits. Our approach integrates tools from communication complexity theory, quantum information-theoretic game design, entanglement-assisted protocol construction, and tight lower-bound analysis. The results resolve long-standing open problems posed by Gavinsky, Gavinsky–Kerenidis–Raz–de Wolf (GKRW), and others. They establish the strongest known entanglement–communication complexity trade-off bounds to date and reveal that infinitesimal changes in entanglement supply can induce exponential improvements in communication efficiency.
📝 Abstract
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 $kge 1$: $Q|^*$ versus $R2^*$: We show that quantum simultaneous protocols with $ ilde{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 $ ilde{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 $ ilde{Theta}(klog 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].
