On the tensorization of the variational distance

If one seeks to estimate the total variation between two product measures $\mathrm{\mid }\mathrm{\mid }{P}_{1:n}^{\otimes }-Q_{1:n}^{\otimes }\mathrm{\mid }\mathrm{\mid }$ in terms of their marginal TV sequence $\delta =(\mathrm{\mid }\mathrm{\mid }{P}_1-Q_1\mathrm{\mid }\mathrm{\mid },\mathrm{\mid }\mathrm{\mid }{P}_2-Q_2\mathrm{\mid }\mathrm{\mid },\dots ,\mathrm{\mid }\mathrm{\mid }{P}_n-Q_n\mathrm{\mid }\mathrm{\mid })$, then trivial upper and lower bounds are provided by$\mathrm{\mid }\mathrm{\mid }\delta \mathrm{\mid }{\mathrm{\mid }}_{\mathrm{\infty }}\le \mathrm{\mid }\mathrm{\mid }{P}_{1:n}^{\otimes }-Q_{1:n}^{\otimes }\mathrm{\mid }\mathrm{\mid }\le \mathrm{\mid }\mathrm{\mid }\delta \mathrm{\mid }{\mathrm{\mid }}_1$. We improve the lower bound to $\mathrm{\mid }\mathrm{\mid }\delta \mathrm{\mid }{\mathrm{\mid }}_2\lesssim \mathrm{\mid }\mathrm{\mid }{P}_{1:n}^{\otimes }-Q_{1:n}^{\otimes }\mathrm{\mid }\mathrm{\mid }$, thereby reducing the gap between the upper and lower bounds from $\sim n$ to \sim\sqrt. Furthermore, we show that {\em any} estimate on $\mathrm{\mid }\mathrm{\mid }{P}_{1:n}^{\otimes }-Q_{1:n}^{\otimes }\mathrm{\mid }\mathrm{\mid }$ expressed in terms of $\delta$ must necessarily exhibit a gap of $\sim \sqrt{n}$ between the upper and lower bounds in the worst case, establishing a sense in which our estimate is optimal. Finally, we identify a natural class of distributions for which $\mathrm{\mid }\mathrm{\mid }\delta \mathrm{\mid }{\mathrm{\mid }}_2$ approximates the TV distance up to absolute multiplicative constants.