Atsuya Hasegawa

Atsuya Hasegawa, François Le Gall, Augusto Modanese

We present relation problems whose input size is $n$ such that they can be solved with no communication for entanglement-assisted quantum communication models, but require $Ω(n)$ qubit communication for $2$-way quantum communication models without prior shared entanglement. This is the maximum separation of quantum communication complexity with and without shared entanglement. To our knowledge, our result even shows the first lower bound on quantum communication complexity without shared entanglement when the upper bound of entanglement-assisted quantum communication models is zero. Our result refutes a quantum analog of Newman's theorem. The problem we consider is parallel repetition of any non-local game which has a perfect quantum strategy and no perfect classical strategy, and for which a parallel repetition theorem holds with exponential decay.

Vahid R. Asadi, Atsuya Hasegawa, François Le Gall

It is known that there exist multi-prover interactive protocols ($\mathsf{MIP}$ protocols) for the complexity class $\mathsf{NEXP}$, succinct $\mathsf{MIP}$ protocols for $\mathsf{NP}$ and multi-prover interactive protocols with shared entanglement ($\mathsf{MIP}^\ast$ protocols) for $\mathsf{RE}$. This extraordinary power of multi-prover interactive proof systems comes from the assumption that provers do not communicate with each other during the protocols. If they are allowed to communicate freely, the setting is the same as in the single-prover case, and the computational power of the system becomes significantly weaker. In this paper, we investigate for the first time the setting where communication (i.e., leakage of information) between provers is allowed but bounded. We introduce two techniques to approach this question and show that multi-prover interactive proof systems are robust against some amount of leakage. Our first technique is based on parallel repetition theorems. We apply it to show that for any polynomial $p$, we can construct two-prover one-round $\mathsf{MIP}$ and $\mathsf{MIP}^\ast$ protocols for $\mathsf{NEXP}$ and $\mathsf{RE}$, respectively, that are robust against $p(n)$ bits of leakage. We further derive our second technique to convert any low-soundness PCP construction to a two-prover one-round $\mathsf{MIP}$ protocol for $\mathsf{NP}$ robust against leakage. We also discuss the relation between robustness against leakage in multi-prover interactive proof systems and the Sliding Scale Conjecture in the PCP literature.