A convolution inequality for entropy over Z2

We prove an inequality for the entropy of a sum of two independent random variables taking values in the group ℤ2. Our inequality is very simply stated, and may be interpreted as a lower bound on the capacity of a cascade of two BSC channels in terms of the capacities of the component BSC channels. The inequality provides an upper bound on the entropy of a sum of two ℤ2-valued random variables, and thus it may also be thought of as a reverse entropy power inequality. One of the intriguing features of this inequality is that it only holds if entropy is measured in bits; i.e., the base with respect to which logarithms are taken matters crucially.