Trade-offs between Entanglement and Communication
Srinivasan Arunachalam, Uma Girish
TL;DR
The paper investigates how entanglement resources influence the advantage of quantum communication over classical approaches, establishing sharp, fine-grained separations for partial function tasks. It develops a unified Fourier-analytic framework and XOR lemmas to bound the power of classical protocols under bounded entanglement and to transfer these bounds to quantum settings, including hybrid models. The two main separations are based on the Forrelation problem (quantum SMP with entanglement vs limited-entanglement classical protocols) and the Boolean Hidden Matching problem (classical SMP with entanglement vs limited-entanglement quantum/one-way/classical models), each supported by explicit XOR-lemma constructions and matrix-hypercontractivity tools. The results advance understanding of entanglement’s role in quantum communication, providing both new separations and techniques applicable to hybrid quantum-classical models and broader resource trade-off questions.
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 $k\ge 1$: $Q\|^*$ versus $R2^*$: We show that quantum simultaneous protocols with $\tildeΘ(k^5 \log^3 n)$ 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 $\tildeΘ(k \log n)$ 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 $\tildeΘ(k\log n)$ 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]. Our techniques can also be used to show advantages of quantum communication models over hybrid classical-quantum models, i.e., models that have a large amount of both classical communication and quantum simultaneous communication.