The classical communication complexity of testing closeness of discrete distributions has recently been studied by Andoni, Malkin and Nosatzki (ICALP'19). In this problem, two players each receive $t$ samples from one distribution over $[n]$, and the goal is to decide whether their two distributions are equal, or are $\epsilon$-far apart in the $l_1$-distance. In the present paper we show that the quantum communication complexity of this problem is $\tilde{O}(n/(t\epsilon^2))$ qubits when the distributions have low $l_2$-norm, which gives a quadratic improvement over the classical communication complexity obtained by Andoni, Malkin and Nosatzki. We also obtain a matching lower bound by using the pattern matrix method. Let us stress that the samples received by each of the parties are classical, and it is only communication between them that is quantum. Our results thus give one setting where quantum protocols overcome classical protocols for a testing problem with purely classical samples.

中文翻译：

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分布式测试的量子通信复杂性

最近，Andoni、Malkin 和 Nosatzki (ICALP'19) 研究了测试离散分布接近度的经典通信复杂性。在这个问题中，两个玩家每人从 $[n]$ 的一个分布中接收 $t$ 个样本，目标是决定他们的两个分布是否相等，或者在 $l_1$-距离。在本文中，我们表明当分布具有低 $l_2$-norm 时，该问题的量子通信复杂度为 $\tilde{O}(n/(t\epsilon^2))$ qubits，这给出了二次改进Andoni、Malkin 和 Nosatzki 获得的经典通信复杂性。我们还通过使用模式矩阵方法获得了一个匹配的下界。让我们强调一下，各方收到的样本都是经典的，只有它们之间的通信才是量子的。因此，我们的结果给出了一种设置，其中量子协议克服了纯经典样本测试问题的经典协议。