  Bipartite entangled Gaussian states provide an important resource for quantum
information based on continuous variables. Perfect complementarity between
one-particle visibility and two-particle visibility in discrete systems, has
been previously generalized to an inequality, rather than equality, for
Gaussian states characterized by varying amounts of squeezing and entanglement.
The latter quantum inequality, however, appears to be due to assessment of
two-particle visibility using an indirect method that first corrects the
two-particle probability distribution by adding and subtracting distributions
with varying degree of entanglement. In this work, we develop a new direct
method for quantifying two-particle visibility based on measurement of a pair
of two-particle observables that commute with the measured pair of
single-particle observables. This direct method treats the two pairs of quantum
observables on equal footing by formally identifying all four observable
distributions as Radon transforms of the original two-particle probability
distribution. The quantum complementarity relation between one-particle
visibility and two-particle visibility obtained through the direct method
deviates from unity by an exponentially small quantity bounded by the amount of
squeezing, which is shown to vanish in the limit of infinite squeezing when the
entangled Gaussian state approaches an ideal Einstein-Podolsky-Rosen state. The
presented results demonstrate the theoretical utility of Radon transforms for
elucidating the nature of two-particle visibility and provide new tools for the
development of quantum applications employing continuous variables.
