  Unitary transformations are routinely modeled and implemented in the field of
quantum optics. In contrast, nonunitary transformations that can involve loss
and gain require a different approach. In this theory work, we present a
universal method to deal with nonunitary networks. An input to the method is an
arbitrary linear transformation matrix of optical modes that does not need to
adhere to bosonic commutation relations. The method constructs a transformation
that includes the network of interest and accounts for full quantum optical
effects related to loss and gain. Furthermore, through a decomposition in terms
of simple building blocks it provides a step-by-step implementation recipe, in
a manner similar to the decomposition by Reck et al. [Reck et al., Phys. Rev.
Lett. 73, 58 (1994)] but applicable to nonunitary transformations. Applications
of the method include the implementation of positive-operator-valued measures
and the design of probabilistic optical quantum information protocols.
