The performance in terms of minimal Bayes’ error probability for detection of ahigh-dimensional random tensor is a fundamental under-studied difficult problem. In this work, weconsider two Signal to Noise Ratio (SNR)-based detection problems of interest. Under the alternativehypothesis, i.e., for a non-zero SNR, the observed signals are either a noisy rank-R tensor admitting aQ-order Canonical Polyadic Decomposition (CPD) with large factors of size Nq R, i.e, for 1 q Q,where R, Nq ! ¥ with R1/q/Nq converge towards a finite constant or a noisy tensor admittingTucKer Decomposition (TKD) of multilinear (M1, . . . ,MQ)-rank with large factors of size Nq Mq,i.e, for 1 q Q, where Nq,Mq ! ¥ with Mq/Nq converge towards a finite constant. The detectionof the random entries (coefficients) of the core tensor in the CPD/TKD is hard to study since theexact derivation of the error probability is mathematically intractable. To circumvent this technicaldifficulty, the Chernoff Upper Bound (CUB) for larger SNR and the Fisher information at low SNRare derived and studied, based on information geometry theory. The tightest CUB is reached forthe value minimizing the error exponent, denoted by s?. In general, due to the asymmetry of thes-divergence, the Bhattacharyya Upper Bound (BUB) (that is, the Chernoff Information calculated ats? = 1/2) can not solve this problem effectively. As a consequence, we rely on a costly numericaloptimization strategy to find s?. However, thanks to powerful random matrix theory tools, a simpleanalytical expression of s? is provided with respect to the Signal to Noise Ratio (SNR) in the twoschemes considered. A main conclusion of this work is that the BUB is the tightest bound at lowSNRs. This property is, however, no longer true for higher SNRs.