The Wehrl entropy is the entropy of the probability distribution in phase space corresponding to the Husimi function in terms of coherent states. We explain the significance of the Wehrl entropy in quantum information theory, and present the theory behind the Lieb conjecture, which states that, in finite dimensions, the minimum Wehrl entropy occurs for Bloch coherent states. This was proven by Lieb and Solovej in 2012. We present the theory behind coherent states, with a particular emphasis on Bloch coherent states, and give a geometrical representation of quantum states as points on a sphere. Using this representation, we identify spherical arrangements of 2–9 points that maximize the Wehrl entropy locally. We conjecture that these maxima are in fact global. Furthermore, we investigate how the maximally entangled symmetric states are related to the states corresponding to maximal Wehrl entropy.
Wehrl used Glauber coherent states to define a map from quantum density matrices to classical phase space densities and conjectured that for Glauber coherent states the mininimum classical entropy would occur for density matrices equal to projectors onto coherent states. This was proved by Lieb in 1978 who also extended the conjecture to Bloch SU(2) spin-coherent states for every angular momentum J. This conjecture is proved here. We also recall our 1991 extension of the Wehrl map to a quantum channel from J to K=J+12,J+1,..., with K=∞ corresponding to the Wehrl map to classical densities. These channels were later recognized as the optimal quantum cloning channels. For each J and J<K⩽∞ we show that the minimal output entropy for the channels occurs for a J coherent state. We also show that coherent states both Glauber and Bloch minimize any concave functional, not just entropy.
We consider the problem of the stability (with sharp exponent) of the Lieb--Solovej inequality for symmetric
SU(N) coherent states, which was obtained only recently by the authors. Here, we propose an elementary proof of this result, based on reformulating the Wehrl-type entropy as a function defined on the unit sphere in
Cd, for some suitable
d, and on some explicit (and somewhat surprising) computations.
Abstract The Husimi function (Q-function) of a quantum state is the distribution function of the density operator in the coherent state representation. It is widely used in theoretical research, such as in quantum optics. The Wehrl entropy is the Shannon entropy of the Husimi function, and is non-zero even for pure states. This entropy has been extensively studied in mathematical physics. Recent research also suggests a significant connection between the Wehrl entropy and many-body quantum entanglement in spin systems. We investigate the statistical interpretation of the Husimi function and the Wehrl entropy, taking the system of N spin-1/2 particles as an example. Due to the completeness of coherent states, the Husimi function and Wehrl entropy can be explained via the positive operator-valued measurement (POVM) theory, although the coherent states are not a set of orthonormal basis. Here, with the help of the Bayes’ theorem, we provide an alternative probabilistic interpretation for the Husimi function and the Wehrl entropy. This interpretation is based on direct measurements of the system, and thus does not require the introduction of an ancillary system as in the POVM theory. Moreover, under this interpretation the classical correspondences of the Husimi function and the Wehrl entropy are just phase-space probability distribution function of N classical tops, and its associated entropy, respectively. Therefore, this explanation contributes to a better understanding of the relationship between the Husimi function, Wehrl entropy, and classical-quantum correspondence. The generalization of this statistical interpretation to continuous-variable systems is also discussed.
Let
PN be the
(N+1)-dimensional Hilbert space of analytic polynomials of degree at most
N. This is the natural environment to define
SU(2) (Bloch) coherent states. Let
Qρ be the Husimi function of a density operator
ρ on
. We prove an isospectral version of Lieb-Solovej inequality: if
ρ↓ is obtained by placing the eigenvalues of
ρ in decreasing order along the monomial basis, then \begin{equation*} \int_{\mathbb{C}}Φ(Q_{ρ}(z))\,dm(z)\leq \int_{\mathbb{C}}Φ(Q_{ρ^{\downarrow }}(z))\,dm(z) \end{equation*}% for every convex function
Φ on
[0,1]. Applying the corresponding reversed inequality to the concave function
Φ(t)=−tlogt gives the Wehrl entropy. In the process it is show that the output state of
ρ under Lieb-Solovej's channel is majorized by the output of the state
ρ↓.
As an application, among all measurable subsets of the sphere having a fixed area, spherical caps maximize every partial sum of the eigenvalues of Toeplitz operators with symbol
1Ω. Equivalently, caps maximize all Ky Fan norms, leading to the sharp inequalities, previously known for
r=1: \begin{equation*} \sum_{j=1}^{r}λ_{j}(Ω) \leq r-\sum_{k=0}^{r-1}(r-k)\binom{N+1}{k} m(Ω)^{k}\bigl(1-m(Ω)\bigr)^{N+1-k}. \end{equation*} This implies isoperimetric inequalities for all Schatten sums of
TΩ, obtained without using the spherical isoperimetric inequality.
We generalize the concept of the Wehrl entropy of quantum states which gives a basis-independent measure of their localization in phase space. We discuss the minimal values and the typical values of these Rényi-Wehrl entropies for pure states for spin systems. According to Lieb's conjecture the minimal values are provided by the spin coherent states. Though Lieb's conjecture remains unproven, we give new proofs of partial results that may be generalized for other systems. We also investigate random pure states and calculate the mean Rényi-Wehrl entropies averaged over the natural measure in the space of pure quantum states.