Trade-Offs Between Entanglement and Communication

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 ≥ 1: Q||* versus R2* : We show that quantum simultaneous protocols with [EQUATION] qubits of entanglement can exponentially outperform two-way randomized protocols with O ( k ) qubits of entanglement. This resolves an open problem from [11] and improves the state-of-the-art separations between quantum simultaneous protocols with entanglement and two-way randomized protocols without entanglement [12, 18]. R||* versus Q||* : We show that classical simultaneous protocols with [EQUATION] qubits of entanglement can exponentially outperform quantum simultaneous protocols with O ( k ) qubits of entanglement, resolving an open question from [15, 12]. The best result prior to our work was a relational separation against protocols without entanglement [15]. R||* versus R1* : We show that classical simultaneous protocols with [EQUATION] 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 [11].