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 by Van Herstraeten and Cerf [Phys. Rev. A 104, 042211 (2021)10.1103/PhysRevA.104.042211] and Hertz et al. [J. Phys. A: Math. Theor. 50, 385301 (2017)10.1088/1751-8121/aa852f]. 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+lnπ. We prove this conjecture for the qubits formed by Fock states |0) and |1) 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 the conjecture's validity. Lastly, we elaborate on the states which are in accordance with our condition.