  We present the quantum theory of coherent Ising machines based on networks of
degenerate optical parametric oscillators (DOPOs).
  In a simple model consisting of two coupled DOPOs, both positive-$P$
representation and truncated Wigner representation predict quantum correlation
and inseparability between the two DOPOs in spite of the open-dissipative
nature of the system.
  Here, we apply the truncated Wigner representation method to coherent Ising
machines with thermal, vacuum, and squeezed reservoir fields. We find that the
probability of finding the ground state of a one-dimensional Ising model
increases substantially as a result of reducing excess thermal noise and
squeezing the incident vacuum fluctuation on the out-coupling port.
