Wehrl entropy

In quantum information theory, the Wehrl entropy,^[1] named after A. Wehrl, is a type of quasi-entropy defined for the Husimi Q representation Q(x,p) ^[2] 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,^[3] 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^[4] 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,^[5] 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).

References

1. Wehrl, A. (1978). "General properties of entropy". Reviews of Modern Physics 50 (2): 221. Bibcode:1978RvMP...50..221W. doi:10.1103/RevModPhys.50.221.
2. Kôdi Husimi (1940). "Some Formal Properties of the Density Matrix", Proc. Phys. Math. Soc. Jpn. 22: 264–314 .
3. Cartwright, N. D. (1975). "A non-negative Wigner-type distribution". Physica A: Statistical Mechanics and its Applications 83: 210–818. doi:10.1016/0378-4371(76)90145-X.
4. C. Zachos, D. Fairlie, and T. Curtright, “Quantum Mechanics in Phase Space” (World Scientific, Singapore, 2005) ISBN 978-981-238-384-6 .
5. Gnutzmann, Sven; Karol Zyczkowski (2001). "Rényi–Wehrl entropies as measures of localization in phase space". J. Phys. A: Math. Gen. 34 (47). arXiv:quant-ph/0106016. Bibcode:2001JPhA...3410123G. doi:10.1088/0305-4470/34/47/317.