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