  Gaussian states are ubiquitous in quantum optics and information processing,
and it is essential to have effective tools for their characterization. One
such tool is a photon-number-resolving detector, and the simplest configuration
involves counting the total number of photons in the state to be characterized.
This motivates the following question: What properties of a multi-mode Gaussian
state are determined by the signal from one detector that measures total number
photons up to some bound? We find that if the Gaussian state occupies $S$ modes
and the probabilities of $n$ photons for all $n\leq 8S$ are known, then we can
determine the spectrum of the Gaussian covariance matrix and the magnitude of
the displacements in each eigenspace of the covariance matrix. Nothing more can
be learned, even if all photon-number probabilities are known. When the state
is pure, the covariance matrix spectrum determines the squeezing parameters of
the state.
