  We explore the role of majorization theory in quantum phase space. To this
purpose, we restrict ourselves to quantum states with positive Wigner functions
and show that the continuous version of majorization theory provides an elegant
and very natural approach to exploring the information-theoretic properties of
Wigner functions in phase space. After identifying all Gaussian pure states as
equivalent in the precise sense of continuous majorization, which can be
understood in light of Hudson's theorem, we conjecture a fundamental
majorization relation: any positive Wigner function is majorized by the Wigner
function of a Gaussian pure state (especially, the bosonic vacuum state or
ground state of the harmonic oscillator). As a consequence, any Schur-concave
function of the Wigner function is lower bounded by the value it takes for the
vacuum state. This implies in turn that the Wigner entropy is lower bounded by
its value for the vacuum state, while the converse is notably not true. Our
main result is then to prove this fundamental majorization relation for a
relevant subset of Wigner-positive quantum states which are mixtures of the
three lowest eigenstates of the harmonic oscillator. Beyond that, the
conjecture is also supported by numerical evidence. We conclude by discussing
some implications of this conjecture in the context of entropic uncertainty
relations in phase space.
