In quantum information theory, the Wehrl entropy,[1] 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 [2] for a comprehensive review of basic properties of classical, quantum and Wehrl entropies, and their implications in statistical mechanics.
The Husimi function[3] is a "classical phase-space" function of position and momentum, and in one dimension is defined for any quantum-mechanical density matrix by
Q\rho(x,p)=\int\phi(x,p|y)*\rho(y,y')\phi(x,p|y')dydy',
\phi(x,p|y)=\pi-1/4\exp(-|y-x|2/2)+ipx).
The Wehrl entropy is then defined as
SW(\rho)=-\intQ\rho(x,p)logQ\rho(x,p)dxdp~.
Such a definition of the entropy relies on the fact that the Husimi Q representation remains non-negative definite,[4] unlike other representations of quantum quasiprobability distributions in phase space. The Wehrl entropy has several important properties:
SW(\rho)\geq0,
SW(\rho)\geq1,
\rho
l{H}=l{H}1 ⊗ l{H}2
SW(\rho1)\leqSW(\rho)
\rho1=Tr2\rho
S(\rho)
SW(\rho)>S(\rho)
SW(\rho)-S(\rho)
| * | |
| S | |
| W(U |
\rhoU) ≠ SW(\rho)
In his original paper Wehrl posted a conjecture that the smallest possible value of Wehrl entropy is 1,
SW(\rho)\geq1,
\rho
x0,p0
\rho0(y,y')=\phi(x0,p
| *\phi(x | |
| 0,p |
0|y')
Soon after the conjecture was posted, E. H. Lieb proved [5] 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 [6] 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.
However, it is not the fully quantum von Neumann entropy in the Husimi representation in phase space, : all the requisite star-products ★ in that entropy have been dropped here. In the Husimi representation, the star products read
\star\equiv\exp\left(
| \hbar | |
| 2 |
({\stackrel{\leftarrow}{\partial}}x-i{\stackrel{\leftarrow}{\partial}}p)({\stackrel{ → }{\partial}}x+i{\stackrel{ → }{\partial}}p)\right)~,
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 dependence (through Q) but not all of it.
Like all entropies, it reflects some measure of non-localization,[8] as the Gauss transform involved in generating 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 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
SU(2)
Consider a space
C2J+1
| J= | 1 |
| 2 |
,1,
| 3 | |
| 2 |
,...
S=(Sx,Sy,Sz)
[Sx,Sy]=iSz
Define
S\pm=Sx\pmiSy
[Sz,S\pm]=\pmS\pm
[S+,S-]=Sz
The eigenstates of
Sz
Sz|s\rangle=s|s\rangle,s=-J,...,J.
For
s=J
|J\rangle\inC2J+1
Sz|J\rangle=J|J\rangle,
S+|J\rangle=0,S-|J\rangle=|J-1\rangle
Denote the unit sphere in three dimensions by
\Xi2=\{\Omega=(\theta,\phi) | 0\leq\theta\leq\pi, 0\leq\phi\leq2\pi\}
L2(\Xi)
| d\Omega= | 2J+1 |
| 4\pi |
\sin\thetad\thetad\phi
The Bloch coherent state is defined by
|\Omega\rangle\equiv\exp\left\{
| 1 | |
| 2 |
\thetaei\phi
| S | ||||
|
\thetae-i\phiS+\right\}|J\rangle
Taking into account the above properties of the state
|J\rangle
|\Omega\rangle=(1+|z|2)-J
| zS- | |
| e |
|J\rangle=(1+|z|2)-J
| J | |
| \sum | |
| M=-J |
zJ-M\binom{2J}{J+M}1/2|M\rangle,
~~z=ei\phi\tan
| \theta | |
| 2 |
|M\rangle=\binom{2J}{J+M}-1/2
| 1 | |
| (J-M)! |
| J-M | |
| S | |
| - |
|J\rangle
Sz
Sz|M\rangle=M|M\rangle
The Bloch coherent state is an eigenstate of the rotated angular momentum operator
Sz
R\theta,\phi=\exp\left\{
| 1 | |
| 2 |
\thetaei\phi
| S | ||||
|
\thetae-i\phiS+\right\}
|\Omega\rangle
R\theta,Sz
| -1 | |
| R | |
| \theta,\phi |
|\Omega\rangle=J|\Omega\rangle
Given a density matrix, define the semi-classical density distribution
\rhocl(\Omega)=\langle\Omega|\rho|\Omega\rangle
\rho
\rhocl
| B(\rho)=S | |
| S | |
| W |
cl(\rhocl)=-\int\rhocl(\Omega) ln\rhocl(\Omega) d\Omega
Scl
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,
| B(\rho)\geq | |
| S | |
| W |
| 2J | |
| 2J+1 |
In 2012 E. H. Lieb and J. P. Solovej proved [9] 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[10] and A. Kulikov, F. Nicola, J. Ortega-Cerda' and P. Tilli.[11]
In E. H. Lieb and J. P. Solovej proved Wehrl's conjecture for Bloch coherent states by generalizing it in the following manner.
For any concave function
f:[0,1] → R
f(x)=-xlogx
\intf(Q\rho(x,p))dxdp\geq\int
| f(Q | |
| \rho0 |
(x,p))dxdp
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
f:[0,1] → R
\intf(\langle\Omega|\rho|\Omega\rangle)d\Omega\geq\intf(|\langle
| 2)d\Omega | |
| \Omega|\Omega | |
| 0\rangle| |
\Omega0\in\Xi2
The uniqueness of the minimizers was proved in the aforementioned papers and.[11]