Wehrl entropy

Consider a space ${\displaystyle \mathbb {C} ^{2J+1}}$ with ${\displaystyle J={\frac {1}{2}},1,{\frac {3}{2}},\dots }$ . We consider a single quantum spin of fixed angular momentum J, and shall denote by ${\displaystyle \mathbf {S} =(S_{x},S_{y},S_{z})}$ the usual angular momentum operators that satisfy the following commutation relations: ${\displaystyle [S_{x},S_{y}]=i\,S_{z}}$ and cyclic permutations.

Define ${\displaystyle S_{\pm }=S_{x}\pm i\,S_{y}}$, then ${\displaystyle [S_{z},S_{\pm }]=\pm S_{\pm }}$ and ${\displaystyle [S_{+},S_{-}]=S_{z}}$.

The eigenstates of ${\displaystyle S_{z}}$ are

${\displaystyle S_{z}|s\rangle =s|s\rangle ,s=-J,\dots ,J.}$

For ${\displaystyle s=J}$ the state ${\displaystyle |J\rangle \in \mathbb {C} ^{2J+1}}$ satisfies: ${\displaystyle S_{z}|J\rangle =J|J\rangle ,}$ and ${\displaystyle S_{+}|J\rangle =0,S_{-}|J\rangle =|J-1\rangle }$.

Denote the unit sphere in three dimensions by

${\displaystyle \Xi _{2}=\{\Omega =(\theta ,\phi )\ |\ 0\leq \theta \leq \pi ,\ 0\leq \phi \leq 2\pi \}}$,

and by ${\displaystyle L^{2}(\Xi )}$ the space of square integrable function on Ξ with the measure

${\displaystyle d\Omega ={\frac {2J+1}{4\pi }}\sin \theta \,d\theta \,d\phi }$.

The Bloch coherent state is defined by

${\displaystyle |\Omega \rangle \equiv \exp \left\{{\frac {1}{2}}\theta e^{i\phi }S_{-}-{\frac {1}{2}}\theta e^{-i\phi }S_{+}\right\}|J\rangle }$.

Taking into account the above properties of the state ${\displaystyle |J\rangle }$, the Bloch coherent state can also be expressed as

${\displaystyle |\Omega \rangle =(1+|z|^{2})^{-J}e^{zS_{-}}|J\rangle =(1+|z|^{2})^{-J}\sum _{M=-J}^{J}z^{J-M}{\binom {2J}{J+M}}^{1/2}|M\rangle ,}$

where ${\displaystyle ~~z=e^{i\phi }\tan {\frac {\theta }{2}}}$, and

${\displaystyle |M\rangle ={\binom {2J}{J+M}}^{-1/2}{\frac {1}{(J-M)!}}\,S_{-}^{J-M}|J\rangle }$

is a normalised eigenstate of ${\displaystyle S_{z}}$ satisfying ${\displaystyle S_{z}|M\rangle =M|M\rangle }$.

The Bloch coherent state is an eigenstate of the rotated angular momentum operator ${\displaystyle S_{z}}$ with a maximum eigenvalue. In other words, for a rotation operator

${\displaystyle R_{\theta ,\phi }=\exp \left\{{\frac {1}{2}}\theta e^{i\phi }S_{-}-{\frac {1}{2}}\theta e^{-i\phi }S_{+}\right\}}$,

the Bloch coherent state ${\displaystyle |\Omega \rangle }$ satisfies

${\displaystyle R_{\theta ,\phi }S_{z}R_{\theta ,\phi }^{-1}\ |\Omega \rangle =J\,|\Omega \rangle }$.

Given a density matrix ρ, define the semi-classical density distribution

${\displaystyle \rho ^{cl}(\Omega )=\langle \Omega |\rho |\Omega \rangle }$.

The Wehrl entropy of ${\displaystyle \rho }$ for Bloch coherent states is defined as a classical entropy of the density distribution ${\displaystyle \rho ^{cl}}$,

${\displaystyle S_{W}^{B}(\rho )=S^{cl}(\rho ^{cl})=-\int \rho ^{cl}(\Omega )\ \ln \rho ^{cl}(\Omega )\ d\Omega }$,

where ${\displaystyle S^{cl}}$ is a classical differential entropy.

The analogue of the Wehrl's conjecture for Bloch coherent states was proposed in [5] in 1978. It suggests the minimum value of the Werhl entropy for Bloch coherent states,

${\displaystyle S_{W}^{B}(\rho )\geq {\frac {2J}{2J+1}}}$,

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 [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 problem of the uniqueness of the minimizer remains unresolved.

For any concave function ${\displaystyle f:[0,1]\rightarrow \mathbb {R} }$ (e.g. ${\displaystyle f(x)=-x\log x}$ as in the definition of the Wehrl entropy), and any density matrix ρ, we have

${\displaystyle \int f(Q_{\rho }(x,p))dx\,dp\geq \int f(Q_{\rho _{0}}(x,p))dx\,dp}$,

where ρ₀ is a pure coherent state defined in the section "Wehrl conjecture".

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 ${\displaystyle f:[0,1]\rightarrow \mathbb {R} }$, and any density matrix ρ we have

${\displaystyle \int f(\langle \Omega |\rho |\Omega \rangle )d\Omega \geq \int f(|\langle \Omega |\Omega _{0}\rangle |^{2})d\Omega }$,

where ${\displaystyle \Omega _{0}\in \Xi _{2}}$ is any point on a sphere.

The uniqueness of the minimizers for either statement remains an open problem.