  For a fixed average energy, the simultaneous estimation of multiple phases
can provide a better total precision than estimating them individually. We show
this for a multimode interferometer with a phase in each mode, using Gaussian
inputs and passive elements, by calculating the covariance matrix. The quantum
Cram\'{e}r-Rao bound provides a lower bound to the covariance matrix via the
quantum Fisher information matrix, whose elements we derive to be the
covariances of the photon numbers across the modes. We prove that this bound
can be saturated. In spite of the Gaussian nature of the problem, the
calculation of non-Gaussian integrals is required, which we accomplish
analytically. We find our simultaneous strategy to yield no more than a
factor-of-2 improvement in total precision, possibly because of a fundamental
performance limitation of Gaussian states. Our work shows that no modal
entanglement is necessary for simultaneous quantum-enhanced estimation of
multiple phases.
