Shannon's entropy power inequality gives a lower bound on the entropy power of the sum of two independent random variables in terms of the individual entropy powers. This statement and some of its consequences are information-theoretic counterparts of certain geometric inequalities. In this talk, I will give an overview of analogous statements for bosonic quantum systems. The first concerns a certain convolution operation between two quantum states: here two independent bosonic modes combine at a beamsplitter. The second involves an operation (originally introduced by Werner) combining a probability distribution on phase space with a quantum state of a bosonic mode. The inequalities have application to certain semigroups as well as capacity problems.

The talk is based on joint work with Stefan Huber, Graeme Smith, and Anna Vershynina.