A direct product theorem for quantum communication complexity with applications to device-independent QKD

We give a direct product theorem for the entanglement-assisted interactive quantum communication complexity of an l-player predicate V. In particular we show that for a distribution p that is product across the input sets of the l players, the success probability of any entanglement-assisted quantum communication protocol for computing n copies of V, whose communication is o(log(eff∗(V, p)) · n), goes down exponentially in n. Here eff∗(V, p) is a distributional version of the quantum efficiency or partition bound introduced by Laplante, Lerays and Roland (2012), which is a lower bound on the distributional quantum communication complexity of computing a single copy of V with respect to p. For a two-input boolean function f , the best result for interactive quantum communication complexity known previously was due to Sherstov (2018), who showed a direct product theorem in terms of the generalized discrepancy, which is a lower bound on communication. Our lower bound on non-distributional communication complexity is in terms of maxproduct p eff ∗(V, p), and there is no known relationship between this and the generalized discrepancy. But we define a distributional version of the generalized discrepancy bound and can show that for a given p, eff∗(V, p) upper bounds it. Moreover, unlike Sherstov’s result, our result works for two-input functions or relations whose outputs are non-boolean as well, and is a strong direct product theorem for functions or relations whose quantum communication complexity is characterized by eff∗(V f , p) for a product p. As an application of our result, we show that it is possible to do device-independent quantum key distribution (DIQKD) without the assumption that devices do not leak any information after inputs are provided to them. We analyze the DIQKD protocol given by Jain, Miller and Shi (2020), and show that when the protocol is carried out with devices that are compatible with n copies of the Magic Square game, it is possible to extract Ω(n) bits of key from it, even in the presence of O(n) bits of leakage. Our security proof is parallel, i.e., the honest parties can enter all their inputs into their devices at once, and works for a leakage model that is arbitrarily interactive, i.e., the devices of the honest parties Alice and Bob can exchange information with each other and with the eavesdropper Eve in any number of rounds, as long as the total number of bits or qubits communicated is bounded. *To appear in FOCS 2021. https://arxiv.org/abs/2106.04299 †Centre for Quantum Technologies and Department of Computer Science, National University of Singapore and MajuLab, UMI 3654, Singapore. Email: rahul@comp.nus.edu.sg ‡Centre for Quantum Technologies, National University of Singapore, Singapore. Email: srijita.kundu@u.nus.edu