  We consider the problem of estimating a rank-one perturbation of a Wigner
matrix in a setting of low signal-to-noise ratio. This serves as a simple model
for principal component analysis in high dimensions. The mutual information per
variable between the spike and the observed matrix, or equivalently, the
normalized Kullback-Leibler divergence between the planted and null models are
known to converge to the so-called {\em replica-symmetric} formula, the
properties of which determine the fundamental limits of estimation in this
model. We provide in this note a short and transparent proof of this formula,
based on simple executions of Gaussian interpolations and standard
concentration-of-measure arguments. The \emph{Franz-Parisi potential}, that is,
the free entropy at a fixed overlap, plays an important role in our proof.
Furthermore, our proof can be generalized straightforwardly to spiked tensor
models of even order.
