Quantum circuit theory of phase-sensitive linear systems

Commutator bracket conservation in linear or linearized systems with loss or gain requires the introduction of operator noise sources. The analogy with thermal noise sources at thermodynamic equilibrium, and with the noise associated with the negative temperature of inverted media leads to theorems for optical noise performance of quantum-mechanical phase-insensitive systems analogous to those developed for classical circuits. These relations are generalized to phase-sensitive systems.