Wigner non-negative states that verify the Wigner entropy conjecture

We present further progress, in the form of analytical results, on the Wigner entropy conjecture set forth in https://link.aps.org/doi/10.1103/PhysRevA.104.042211 and https://iopscience.iop.org/article/10.1088/1751-8121/aa852f/meta. Said conjecture asserts that the differential entropy defined for non-negative, yet physical, Wigner functions is minimized by pure Gaussian states while the minimum entropy is equal to $1+\ln\pi$. We prove this conjecture for the qubits formed by Fock states $|0\rangle$ and $|1\rangle$ that correspond to non-negative Wigner functions. In particular, we derive an explicit form of the Wigner entropy for those states lying on the boundary of the set of Wigner non-negative qubits. We then consider general mixed states and derive a sufficient condition for Wigner non-negativity. For states satisfying our condition we verify that the conjecture is true. Lastly, we elaborate on the states of the set which is in accordance with our condition.