Quantum Wigner entropy

We define the Wigner entropy of a quantum state as differential Shannon function state. This quantity is properly defined only for states that possess positive function, which we name Wigner-positive states, but argue it proper measure uncertainty in phase space. It invariant under symplectic transformations (displacements, rotations, and squeezing) conjecture lower bounded by $ln\ensuremath{\pi}+1$ within convex set states. reaches this bound Gaussian pure are natural minimum-uncertainty bears resemblance with Wehrl-Lieb conjecture, prove over subset passive harmonic oscillator particular relevance thermodynamics. Along way, present simple technique to build broad class exploiting an optical beam splitter reveal unexpectedly decomposition extremal The anticipated be significant physical quantity, example, optics where allows us establish entropy-power inequality. also opens way towards stronger entropic relations. Finally, Wigner-R\'enyi extended reached