We show that computing the total variation distance between two product distributions is $\#\mathsf{P}$-complete. This is in stark contrast with other distance measures such as Kullback-Leibler, Chi-square, and Hellinger, which tensorize over the marginals leading to efficient algorithms.

翻译：我们证明计算两个乘积分布之间的总变差距离是$\#\mathsf{P}$-完全的。这与Kullback-Leibler散度、卡方距离和Hellinger距离等其他度量形成鲜明对比，这些度量在边缘分布上具有张量积性质，从而可借助高效算法求解。