Wehrl entropy

In quantum information theory, the Wehrl entropy, named after A. Wehrl, is a type of quasi-entropy defined for the Husimi Q representation Q(x,p) of the phase-space quasiprobability distribution. It is defined as

$S_{Q}=-\int Q(x,p)\log Q(x,p)\,dx\,dp~.$

Such a definition of entropy relies on the fact that the Husimi Q representation remains non-negative definite, unlike other representations of quantum quasiprobability distributions in phase space.

However, it is not the fully quantum von Neumann entropy in the Husimi representation in phase space, − ∫ Q ★ log_★Q  dx dp: all the requisite star-products ★ in that entropy have been dropped here. In the Husimi representation, the star products read

$\star \equiv \exp \left({\frac {\hbar }{2}}({\stackrel {\leftarrow }{\partial }}_{x}-i{\stackrel {\leftarrow }{\partial }}_{p})({\stackrel {\rightarrow }{\partial }}_{x}+i{\stackrel {\rightarrow }{\partial }}_{p})\right)~,$

and are isomorphic to the Moyal products of the Wigner–Weyl representation.

The Wehrl entropy, then, may be thought of as a type of heuristic semiclassical approximation to the full quantum von Neumann entropy, since it retains some ħ dependence (through Q) but not all of it.

Like all entropies, it reflects some measure of non-localization, as the Gauss transform involved in generating Q and the sacrifice of the star operators have effectively discarded information. In general, for the same state, the Wehrl entropy exceeds the von Neumann entropy (which vanishes for pure states).