  We show that the positivity of the Wigner function of Gaussian states and
measurements provides an elegant way to bound the discriminating power of
"linear optics", which we formalise as Gaussian measurement operations
augmented by classical (feed-forward) communication (GOCC). This allows us to
reproduce and generalise the result of Takeoka and Sasaki [PRA 78:022320,
2008], which tightly characterises the GOCC norm distance of coherent states,
separating it from the optimal distinguishability according to Helstrom's
theorem.
  Furthermore, invoking ideas from classical and quantum Shannon theory we show
that there are states, each a probabilistic mixture of multi-mode coherent
states, which are exponentially reliably discriminated in principle, but appear
exponentially close judging from the output of GOCC measurements. In analogy to
LOCC data hiding, which shows an irreversibility in the preparation and
discrimination of states by the restricted class of local operations and
classical communication (LOCC), we call the present effect GOCC data hiding.
  We also present general bounds in the opposite direction, guaranteeing a
minimum of distinguishability under measurements with positive Wigner function,
for any bounded-energy states that are Helstrom distinguishable. We conjecture
that a similar bound holds for GOCC measurements.
