Wehrl entropy

The Husimi function{{cite journal|author=Kôdi Husimi|year=1940|title=Some Formal Properties of the Density Matrix|journal=Proceedings of the Physico-Mathematical Society of Japan|series=3|volume=22|issue=4|pages=264–314 |doi=10.11429/ppmsj1919.22.4_264|doi-access=free}} is a "classical phase-space" function of position {{mvar|x}} and momentum {{mvar|p}}, and in one dimension is defined for any quantum-mechanical density matrix {{mvar|ρ }} by

:$Q\_\backslash rho(x,p)=\backslash int\; \backslash phi(x,p\; |\; y)^*\; \backslash rho\; (y,\; y\text{'})\backslash phi\; (x,p|y\text{'})dy\; dy\text{'},$

where {{mvar|φ}} is a "(Glauber) coherent state", given by

:$\backslash phi(x,p|y)=\backslash pi^\{-1/4\}\backslash exp(-|y-x|^2/2)+i\backslash ,\; px).$

(It can be understood as the Weierstrass transform of the Wigner quasi-probability distribution.)

The Wehrl entropy is then defined as

: $S\_W(\backslash rho)\; =\; -\backslash int\; Q\_\backslash rho(x,p)\; \backslash log\; Q\_\backslash rho(x,p)\; \backslash ,\; dx\; \backslash ,\; dp\; ~.$

The definition can be easily generalized to any finite dimension.

Such a definition of the entropy relies on the fact that the Husimi Q representation remains non-negative definite,{{Cite journal|year=1975|title=A non-negative Wigner-type distribution|journal=Physica A: Statistical Mechanics and Its Applications|volume=83|pages=210–818|bibcode=1976PhyA...83..210C|doi=10.1016/0378-4371(76)90145-X|last1=Cartwright|first1=N. D.|issue=1 }} unlike other representations of quantum quasiprobability distributions in phase space. The Wehrl entropy has several important properties:

1. It is always positive, $S\_W(\backslash rho)\backslash geq\; 0,$ like the full quantum von Neumann entropy, but unlike the classical differential entropy which can be negative at low temperature. In fact, the minimum value of the Wehrl entropy is 1, i.e. $S\_W(\backslash rho)\backslash geq\; 1,$ as discussed below in the section "Werhl's conjecture".
2. The entropy for the tensor product of two systems is always greater than the entropy of one system. In other words, for a state $\backslash rho$ on a Hilbert space $\backslash mathcal\{H\}=\backslash mathcal\{H\}\_1\backslash otimes\backslash mathcal\{H\}\_2$, we have $S\_W(\backslash rho\_1)\backslash leq\; S\_W(\backslash rho)$, where $\backslash rho\_1=\backslash mathrm\{Tr\}\_2\backslash ,\; \backslash rho$. Note that the quantum von Neumann entropy, $S(\backslash rho)$, does not have this property, as can be clearly seen for a pure maximally entangled state.
3. The Wehrl entropy is strictly lower bounded by a von Neumann entropy, $S\_W(\backslash rho)\; >\; S(\backslash rho)$. There is no known upper or lower bound (other than zero) for the difference $S\_W(\backslash rho)-S(\backslash rho)$.
4. The Wehrl entropy is not invariant under all unitary transformations, unlike the von Neumann entropy. In other words, $S\_W(U^*\; \backslash rho\; \backslash ,U)\backslash neq\; S\_W(\backslash rho)$ for a general unitary {{mvar|U}}. It is, however, invariant under certain unitary transformations.

In his original paper Wehrl posted a conjecture that the smallest possible value of Wehrl entropy is 1, $S\_W(\backslash rho)\backslash geq\; 1,$ and it occurs if and only if the density matrix $\backslash rho$ is a pure state projector onto any coherent state, i.e. for all choices of $x\_0,\; p\_0$,

:$\backslash rho\_0(y,\; y\text{'})=\backslash phi(x\_0,p\_0|y)^*\backslash phi(x\_0,p\_0|y\text{'})$.

Soon after the conjecture was posted, E. H. Lieb proved {{cite journal | last=Lieb | first=Elliott H. | title=Proof of an entropy conjecture of Wehrl | journal=Communications in Mathematical Physics | publisher=Springer Science and Business Media LLC | volume=62 | issue=1 | year=1978 | issn=0010-3616 | doi=10.1007/bf01940328 | pages=35–41 | bibcode=1978CMaPh..62...35L| s2cid=189836756 | url=http://projecteuclid.org/euclid.cmp/1103904300 }} 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 {{Cite journal|last=Carlen|first=E.|year=1991|title=Some integral identities and inequalities for entire functions and their application to the coherent state transform|journal=Journal of Functional Analysis|volume=97|pages=231–249|doi=10.1016/0022-1236(91)90022-W|doi-access=free}} 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, {{math|− ∫ 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

: $\backslash star\; \backslash equiv\; \backslash exp\backslash left(\; \backslash frac\{\backslash hbar\}\{2\}$

({\stackrel{\leftarrow}{\partial}}_x -i {\stackrel{\leftarrow}{\partial}}_p) ({\stackrel{\rightarrow}{\partial}}_x + i{\stackrel{\rightarrow}{\partial}}_p ) \right)~,

and are isomorphicC. Zachos, D. Fairlie, and T. Curtright, “Quantum Mechanics in Phase Space” (World Scientific, Singapore, 2005) {{ISBN|978-981-238-384-6}} . 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 {{mvar|ħ}} dependence (through Q) but not all of it.

Like all entropies, it reflects some measure of non-localization,{{cite journal|last=Gnutzmann|first=Sven|year=2001|title=Rényi–Wehrl entropies as measures of localization in phase space|journal=J. Phys. A: Math. Gen.|volume=34|issue=47|pages=10123|arxiv=quant-ph/0106016|bibcode=2001JPhA...3410123G|doi=10.1088/0305-4470/34/47/317|author2=Karol Zyczkowski|s2cid=7327137|author-link2=Karol Życzkowski}}

as the Gauss transform involved in generating {{mvar|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 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 $SU(2)$ for quantum spin systems.

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

Define $S\_\backslash pm=S\_x\backslash pm\; i\backslash ,\; S\_y$, then $[S\_z,\; S\_\backslash pm]=\backslash pm\; S\_\backslash pm$ and $[S\_+,\; S\_-]=S\_z$.

The eigenstates of $S\_z$ are

:$S\_z|s\backslash rangle=s|s\backslash rangle,\; s=-J,\backslash dots,\; J.$

For $s=J$ the state $|J\backslash rangle\backslash in\; \backslash mathbb\{C\}^\{2J+1\}$ satisfies: $S\_z|J\backslash rangle=J|J\backslash rangle,$ and $S\_+|J\backslash rangle=0,\; S\_-|J\backslash rangle=|J-1\backslash rangle$.

Denote the unit sphere in three dimensions by

:$\backslash Xi\_2=\backslash \{\backslash Omega=(\backslash theta,\; \backslash phi)\backslash \; |\; \backslash \; 0\backslash leq\; \backslash theta\; \backslash leq\; \backslash pi,\backslash \; 0\backslash leq\; \backslash phi\backslash leq\; 2\backslash pi\backslash \}$,

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

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

The Bloch coherent state is defined by

:$|\backslash Omega\backslash rangle\backslash equiv\; \backslash exp\backslash left\backslash \{\backslash frac\{1\}\{2\}\backslash theta\; e^\{i\backslash phi\}S\_--\backslash frac\{1\}\{2\}\backslash theta\; e^\{-i\backslash phi\}S\_+\backslash right\backslash \}|J\backslash rangle$.

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

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

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

:$|M\backslash rangle=\backslash binom\{2J\}\{J+M\}^\{-1/2\}\backslash frac\{1\}\{(J-M)!\}\backslash ,\; S\_-^\{J-M\}|J\backslash rangle$

is a normalised eigenstate of $S\_z$ satisfying $S\_z|M\backslash rangle=M|M\backslash rangle$.

The Bloch coherent state is an eigenstate of the rotated angular momentum operator $S\_z$ with a maximum eigenvalue. In other words, for a rotation operator

:$R\_\{\backslash theta,\backslash phi\}=\backslash exp\backslash left\backslash \{\backslash frac\{1\}\{2\}\backslash theta\; e^\{i\backslash phi\}\; S\_--\backslash frac\{1\}\{2\}\backslash theta\; e^\{-i\backslash phi\}\; S\_+\; \backslash right\backslash \}$,

the Bloch coherent state $|\backslash Omega\backslash rangle$ satisfies

:$R\_\{\backslash theta,\; \backslash phi\}\; S\_z\; R^\{-1\}\_\{\backslash theta,\; \backslash phi\}\; \backslash \; |\backslash Omega\backslash rangle=J\backslash ,|\backslash Omega\backslash rangle$.

Given a density matrix {{mvar|ρ}}, define the semi-classical density distribution

:$\backslash rho^\{cl\}(\backslash Omega)=\backslash langle\; \backslash Omega|\; \backslash rho\; |\backslash Omega\; \backslash rangle$.

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

:$S\_W^B(\backslash rho)=S^\{cl\}(\backslash rho^\{cl\})=-\backslash int\; \backslash rho^\{cl\}(\backslash Omega)\backslash \; \backslash ln\; \backslash rho^\{cl\}(\backslash Omega)\backslash \; d\backslash Omega$,

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

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,

:$S\_W^B(\backslash rho)\backslash geq\; \backslash frac\{2J\}\{2J+1\}$,

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 {{Cite journal|last1=Lieb|first1=E.H.|last2=Solovej|first2=J.P.|year=2014|title=Proof of an entropy conjecture for Bloch coherent spin states and its generalizations|journal=Acta Mathematica|volume=212|issue=2|pages=379–398|doi=10.1007/s11511-014-0113-6|arxiv=1208.3632|s2cid=119166106}} 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{{cite journal |last1=Frank |first1=R.L. |title=Sharp inequalities for coherent states and their optimizers |journal=Advanced Nonlinear Studies |date=2023 |volume=23 |issue=1 |page=Paper No. 20220050, 28 |doi=10.1515/ans-2022-0050 |url=https://doi.org/10.1515/ans-2022-0050|arxiv=2210.14798 }} and A. Kulikov, F. Nicola, J. Ortega-Cerda' and P. Tilli.{{cite arXiv |last1=Kulikov |first1=A. |last2=Nicola |first2=F. |last3=Ortega-Cerda' |first3=J. |last4=Tilli |first4=P. |title=A monotonicity theorem for subharmonic functions on manifolds |date=2022 |class=math.CA |eprint=2212.14008}}

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

:$\backslash int\; f(Q\_\backslash rho(x,p))dx\backslash ,\; dp\; \backslash geq\; \backslash int\; f(Q\_\{\backslash rho\_0\}(x,p))dx\backslash ,\; dp$,

where {{mvar|ρ}}₀ 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 $f:\; [0,1]\backslash rightarrow\; \backslash mathbb\{R\}$, and any density matrix {{mvar|ρ}} we have

:$\backslash int\; f(\backslash langle\; \backslash Omega|\backslash rho|\backslash Omega\backslash rangle)d\backslash Omega\; \backslash geq\; \backslash int\; f(|\backslash langle\; \backslash Omega|\backslash Omega\_0\backslash rangle|^2)d\backslash Omega$,

where $\backslash Omega\_0\backslash in\backslash Xi\_2$ is any point on a sphere.

The uniqueness of the minimizers was proved in the aforementioned papers and.

• Coherent state
• Entropy
• Information theory and measure theory
• Lieb conjecture
• Quantum information
• Quantum mechanics
• Spin
• Statistical mechanics
• Von Neumann entropy

Category:Quantum mechanical entropy

Category:Mathematical physics