The classical Entropy Power Inequality (EPI) bounds the Shannon
entropy of the sum of random variables, and is at the basis of
information theory and probability. Its proof is based on the heat
semigroup and on the so-called Stam inequality on the Fisher
information of a convolution. The setup of the EPI can represent the
addition of Gaussian noise to a given signal, so this inequality has
been crucial for the determination of the capacity of various Gaussian
communication channels.
However, the electromagnetic field used in radio communications is
ultimately a quantum-mechanical entity.
We prove a quantum generalization the EPI (qEPI), that permits to
extend to the quantum setting all the results on channel capacities
obtained with its classical counterpart. Our proof is based on the
quantum version of the heat semigroup and on a quantum generalization
of the Stam inequality.
Our qEPI supports the validity of the so-called Entropy Photon Number
Inequality, the last of a series of entropic inequalities associated
with Bosonic systems still left unsolved.