  To quantify quantum optical coherence requires both the particle- and
wave-natures of light. For an ideal laser beam [1,2,3], it can be thought of
roughly as the number of photons emitted consecutively into the beam with the
same phase. This number, $\mathfrak{C}$, can be much larger than $\mu$, the
number of photons in the laser itself. The limit on $\mathfrak{C}$ for an ideal
laser was thought to be of order $\mu^2$ [4,5]. Here, assuming nothing about
the laser operation, only that it produces a beam with certain properties close
to those of an ideal laser beam, and that it does not have external sources of
coherence, we derive an upper bound: $\mathfrak{C} = O(\mu^4)$. Moreover, using
the matrix product states (MPSs) method [6,7,8,9], we find a model that
achieves this scaling, and show that it could in principle be realised using
circuit quantum electrodynamics (QED) [10]. Thus $\mathfrak{C} = O(\mu^2)$ is
only a standard quantum limit (SQL); the ultimate quantum limit, or Heisenberg
limit, is quadratically better.
