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arXiv:1208.3632 · Acta Mathematica

Proof of an entropy conjecture for Bloch coherent spin states and its generalizations

Élliott H. Lieb, Jan Philip Solovej

Published Jan 1, 2014• 61 citations

Abstract

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.

DOI: 10.1007/s11511-014-0113-6

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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.

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.

Let

G

be a compact connected semisimple Lie group, and let

π:G\to U(V_λ)

be a finite-dimensional irreducible unitary representation associated with the highest weight

λ

. We prove that coherent projectors are the unique minimizers of the Wehrl entropy over all density matrices on

V_λ

. Similar extremal properties hold for power functions in place of the entropy function. The proof uses a second variation at the extremizer by perturbation in directions associated with Killing fields, similar to the strategy of Frank and Lieb used in \cite{FrankLieb}. Then the problem reduces to the uncertainty principle characterizing coherent states. This solves the Wehrl-type entropy problem for arbitrary compact connected semisimple Lie groups.

Let

\mathcal{P}_{N}

be the

(N+1)

-dimensional Hilbert space of analytic polynomials of degree at most

V_λ

. This is the natural environment to define

SU(2)

(Bloch) coherent states. Let

Q_{ρ}

be the Husimi function of a density operator

ρ

on

% \mathcal{P}_{N}

. We prove an isospectral version of Lieb-Solovej inequality: if

ρ^{\downarrow }

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)=-t\log t

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^{\downarrow }

. 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

1_Ω

. 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

T_Ω

, obtained without using the spherical isoperimetric inequality.

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

\mathbb{C}^d

, for some suitable

d

, and on some explicit (and somewhat surprising) computations.