  We point out that area laws of quantum-information concepts indicate
limitations of block transformations as well-behaved real-space renormalization
group (RG) maps, which in turn guides the design of better RG schemes.
Mutual-information area laws imply the difficulty of Kadanoff's block-spin
method in two dimensions (2D) or higher due to the growth of short-scale
correlations among the spins on the boundary of a block. A leap to the
tensor-network RG, in hindsight, follows the guidance of mutual information and
is efficient in 2D, thanks to its mixture of quantum and classical perspectives
and the saturation of entanglement entropy in 2D. In three dimensions (3D),
however, entanglement grows according to the area law, posing a threat to 3D
block-tensor map as an apt RG transformation. As a numerical evidence, we show
that estimations of 3D Ising critical exponents fail to improve by retaining
more couplings. As a guidance to proceed, a tensor-network toy model is
proposed to capture the 3D entanglement-entropy area law.
