Wehrl entropy

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Wehrl entropy is a simple way to measure how spread out a quantum state is in phase space. It uses the Husimi Q representation, a nonnegative, smoothed version of the quantum state written as a probability-like distribution over position and momentum (or over other phase-space coordinates).

How it works in plain terms:
- For a quantum state described by a density matrix, you look at its overlap with all coherent states. This overlap gives you the Husimi Q function, a smooth “classical” distribution that always stays nonnegative.
- The Wehrl entropy is the usual classical entropy of this Q function. In other words, you compute how uncertain or spread out the Q distribution is.

Two important facts about Wehrl entropy:
- A fundamental result called Wehrl’s conjecture, later proven, says the smallest possible value is 1. Equality happens precisely when the quantum state is a pure coherent state (a highly classical-like state).
- In general, the Wehrl entropy is larger than the quantum von Neumann entropy. The von Neumann entropy vanishes for pure states, but the Wehrl entropy does not necessarily vanish, because it is a more “classical” measure that loses some quantum details.

Extensions and generalizations:
- The idea of Wehrl entropy can be applied not only to the usual Glauber (harmonic-oscillator) coherent states but also to other kinds, such as Bloch (SU(2)) coherent states used for quantum spins.
- For spins, the Bloch-coherent-state version has a similar minimum: the lowest Wehrl entropy is achieved by pure Bloch coherent states. This result was substantially developed in the 2010s, with a complete uniqueness proof published in 2022.
- More generally, when you replace the entropy function with any concave function, you get a broad family of related statements. The minimum is still reached for pure coherent states, and the minimizers are unique in the cases proved so far.

Bottom line:
Wehrl entropy provides a simple, intuitive link between quantum states and their classical-like localization in phase space. It captures how spread out a state is in a way that is easy to interpret, while acting as a semiclassical counterpart to the full quantum von Neumann entropy.