  We theoretically propose and experimentally demonstrate a nonclassicality
test of single-mode field in phase space, which has an analogy with the
nonlocality test proposed by Banaszek and Wodkiewicz [Phys. Rev. Lett. 82, 2009
(1999)]. Our approach to deriving the classical bound draws on the fact that
the Wigner function of a coherent state is a product of two independent
distributions as if the orthogonal quadratures (position and momentum) in phase
space behave as local realistic variables. Our method detects every pure
nonclassical Gaussian state, which can also be extended to mixed states.
Furthermore, it sets a bound for all Gaussian states and their mixtures,
thereby providing a criterion to detect a genuine quantum non-Gaussian state.
Remarkably, our phase-space approach with invariance under Gaussian unitary
operations leads to an optimized test for a given non-Gaussian state. We
experimentally show how this enhanced method can manifest quantum
non-Gaussianity of a state by simply choosing phase-space points appropriately,
which is essentially equivalent to implementing a squeezing operation on a
given state.
