quantum field theory - Mass gap for photons

I am puzzled by the answers to the question:

What is a mass gap?

There, Ron Maimon's answer gives a clear-cut definition, which I suppose applies to any quantum field theory with Hamiltonian $H$, that the theory has a mass gap if there is a positive constant $A$ such that $$\langle \psi| H |\psi \rangle\geq \langle 0 |H | 0 \rangle +A$$ for all nonzero (normalized) $\psi$.

But then, Arnold Neumaier says

QED has no mass gap, as observable photons are massless states.

I would quite appreciate a brief explanation of this statement. The definition is concerned with the minimum possible energy for non-zero states. So I don't see why the photons having zero mass would imply the absence of a mass gap.