Classic entropy of a quantum-mechanical density matrix
In quantum information theory, the Wehrl entropy, named after Alfred Wehrl, is a classical entropy of a quantum-mechanical density matrix. It is a type of quasi-entropy defined for the Husimi Q representation of the phase-space quasiprobability distribution. See for a comprehensive review of basic properties of classical, quantum and Wehrl entropies, and their implications in statistical mechanics.
Definitions.
The Husimi function is a "classical phase-space" function of position x and momentum p, and in one dimension is defined for any quantum-mechanical density matrix ρ by 
formula_1
where φ is a "(Glauber) coherent state", given by
formula_2
The Wehrl entropy is then defined as
 formula_3
The definition can be easily generalized to any finite dimension.
Properties.
Such a definition of the entropy relies on the fact that the Husimi Q representation remains non-negative definite, unlike other representations of quantum quasiprobability distributions in phase space. The Wehrl entropy has several important properties: 
Wehrl's conjecture.
In his original paper Wehrl posted a conjecture that the smallest possible value of Wehrl entropy is 1, formula_5 and it occurs if and only if the density matrix formula_6 is a pure state projector onto any coherent state, i.e. for all choices of formula_16,
formula_17.
Soon after the conjecture was posted, E. H. Lieb proved that the minimum of the Wehrl entropy is 1, and it occurs when the state is a projector onto any coherent state.
In 1991 E. Carlen proved the uniqueness of the minimizer, i.e. the minimum of the Wehrl entropy occurs only when the state is a projector onto any coherent state.
The analog of the Wehrl conjecture for systems with a classical phase space isomorphic to the sphere (rather than the plane) is the Lieb conjecture.
Discussion.
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
 formula_18
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, as indicated, for the same state, the Wehrl entropy exceeds the von Neumann entropy (which vanishes for pure states).
Wehrl entropy for Bloch coherent states.
Wehrl entropy can be defined for other kinds of coherent states. For example, it can be defined for Bloch coherent states, that is, for angular momentum representations of the group formula_19 for quantum spin systems.
Bloch coherent states.
Consider a space formula_20 with formula_21 . We consider a single quantum spin of fixed angular momentum J, and shall denote by formula_22 the usual angular momentum operators that satisfy the following commutation relations: formula_23 and cyclic permutations.
Define formula_24, then formula_25 and formula_26.
The eigenstates of formula_27 are
formula_28
For formula_29 the state formula_30 satisfies: formula_31 and formula_32.
Denote the unit sphere in three dimensions by 
formula_33, 
and by formula_34 the space of square integrable function on Ξ with the measure 
formula_35.
The Bloch coherent state is defined by 
formula_36.
Taking into account the above properties of the state formula_37, the Bloch coherent state can also be expressed as
formula_38
where formula_39, and 
formula_40 
is a normalised eigenstate of formula_27 satisfying formula_42.
The Bloch coherent state is an eigenstate of the rotated angular momentum operator formula_27 with a maximum eigenvalue. In other words, for a rotation operator 
formula_44, 
the Bloch coherent state formula_45 satisfies 
formula_46.
Wehrl entropy for Bloch coherent states.
Given a density matrix ρ, define the semi-classical density distribution 
formula_47. 
The Wehrl entropy of formula_6 for Bloch coherent states is defined as a classical entropy of the density distribution formula_49,
formula_50,
where formula_51 is a classical differential entropy.
Wehrl's conjecture for Bloch coherent states.
The analogue of the Wehrl's conjecture for Bloch coherent states was proposed in in 1978. It suggests the minimum value of the Werhl entropy for Bloch coherent states, 
formula_52, 
and states that the minimum is reached if and only if the state is a pure Bloch coherent state.
In 2012 E. H. Lieb and J. P. Solovej proved a substantial part of this conjecture, confirming the minimum value of the Wehrl entropy for Bloch coherent states, and the fact that it is reached for any pure Bloch coherent state. The uniqueness of the minimizers was proved in 2022 by R. L. Frank and A. Kulikov, F. Nicola, J. Ortega-Cerda' and P. Tilli.
Generalized Wehrl's conjecture.
In E. H. Lieb and J. P. Solovej proved Wehrl's conjecture for Bloch coherent states by generalizing it in the following manner.
Generalized Wehrl's conjecture.
For any concave function formula_53 (e.g. formula_54 as in the definition of the Wehrl entropy), and any density matrix ρ, we have
formula_55,
where ρ0 is a pure coherent state defined in the section "Wehrl conjecture".
Generalized Wehrl's conjecture for Bloch coherent states.
Generalized Wehrl's conjecture for Glauber coherent states was proved as a consequence of the similar statement for Bloch coherent states. For any concave function formula_53, and any density matrix ρ we have
formula_57,
where formula_58 is any point on a sphere.
The uniqueness of the minimizers was proved in the aforementioned papers and.