This paper gives a pleasing general condition for the "all or nothing" phenemonon around the SNR for signal recovery in additive Gaussian models. Specifically, this paper shows that sparse tensor PCA exhibits he "all or nothing" phenomenon. That is, there's a threshold SNR below which, nothing of the sparse added spike can be recovered while above which, almost everything can be recovered. Beyond the actual results, the paper gives a simple and intuitively interpretable sufficient condition based on the KL divergence between two distributions that governs the all or nothing phenomenon. This paper studies an important class of statistical signal recovery models and gives an elegant insight into when and what kind of recovery is possible. I am pleased to recommend acceptance to NeurIPS.