  We present a characterization of topological phases in photonic lattices. Our
theory relies on a formal equivalence between the singular value decomposition
of the non-Hermitian coupling matrix and the diagonalization of an effective
Hamiltonian. By means of that mapping we unveil an application of topological
band theory to the description of quantum amplification with non-reciprocal
systems. We exemplify our ideas with an array of photonic cavities which can be
mapped into a topological insulator Hamiltonian in the AIII symmetry class. We
investigate stability properties and prove the existence of stable
topologically non-trivial steady-state phases. Finally, we show numerically
that the topological amplification process is robust against disorder in the
lattice parameters.
