Continuous majorization in quantum phase space

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