Proof of the Wehrl-type Entropy Conjecture for Symmetric $${SU(N)}$$ Coherent States

Lieb, Élliott H.; Solovej, Jan Philip

doi:10.1007/s00220-016-2596-9

Cited by 12 publications

(9 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It was first shown for low spin in [29] and then finally in general in [35,36]. Computational evidence suggested in fact an analogous but much stronger conjecture for any concave function φ(x) [29,37], which has also been settled affirmatively in [36]. For our application, the maximal value of the Wehrl entropy is more interesting.…”

Section: A Coherent States Multipole Vectors and Entropysupporting

confidence: 54%

“…This fact is surprisingly difficult to prove. It was first shown for low spin in [29] and then finally in general in [35,36]. Computational evidence suggested in fact an analogous but much stronger conjecture for any concave function φ(x) [29,37], which has also been settled affirmatively in [36].…”

Section: A Coherent States Multipole Vectors and Entropymentioning

confidence: 89%

“…Finally, our work is also interesting from a mathematical perspective. The fact that the Wehrl entropy is minimized in general by SU(N)-coherent states is known as the Lieb conjecture and has been proven [36]. The same question is, however, still open for the angular entropy.…”

Section: Summary and Discussionmentioning

confidence: 99%

See 2 more Smart Citations

Entropy methods for CMB analysis of anisotropy and non-Gaussianity

2019

View full text Add to dashboard Cite

In recent years, high-resolution cosmic microwave background (CMB) measurements have opened up the possibility to explore statistical features of the temperature fluctuations down to very small angular scales. One method that has been used is the Wehrl entropy, which is, however, extremely costly in terms of computational time. Here, we propose several different pseudoentropy measures (projection, angular, and quadratic) that agree well with the Wehrl entropy, but are significantly faster to compute. All of the presented alternatives are rotationally invariant measures of entanglement after identifying each multipole l of temperature fluctuations with a spin-l quantum state and are very sensitive to non-Gaussianity, anisotropy, and statistical dependence of spherical harmonic coefficients in the data. We provide a simple proof that the projection pseudoentropy converges to the Wehrl entropy with increasing dimensionality of the ancilla projection space. Furthermore, for l = 2, we show that both the Wehrl entropy and the angular pseudoentropy can be expressed as one-dimensional functions of the squared chordal distance of multipole vectors, giving a tight connection between the two measures. We also show that the angular pseudoentropy can clearly distinguish between Gaussian and non-Gaussian temperature fluctuations at large multipoles and henceforth provides a non-brute-force method for identifying non-Gaussianities. This allows us to study possible hints of statistical anisotropy and non-Gaussianity in the CMB up to multipole l = 1000 using Planck 2015, Planck 2018, and WMAP 7-yr full sky data. We find that l = 5 and l = 28 have a large entropy at 2-3σ significance and a slight hint towards a connection of this with the cosmic dipole. On a wider range of large angular scales we do not find indications of violation of isotropy or Gaussianity. We also find a small-scale range, l ∈ [895, 905], that is incompatible with the assumptions at about 3σ level, although how much this significance can be reduced by taking into account the selection effect, i.e.,, how likely it is to find ranges of a certain size with the observed features, and inhomogeneous noise is left as an open question. Furthermore, we find overall similar results in our analysis of the 2015 and the 2018 data. Finally, we also demonstrate how a range of angular momenta can be studied with the range angular pseudoentropy, which measures averages and correlations of different multipoles. Our main purpose in this work is to introduce the methods, analyze their mathematical background, and demonstrate their usage for providing researchers in this field with an additional tool. We believe that the formalism developed here can underpin future studies of the Gaussianity and isotropy of the CMB and help to identify deviations, especially at small angular scales.

show abstract

Section: A Coherent States Multipole Vectors and Entropysupporting

confidence: 54%

Section: A Coherent States Multipole Vectors and Entropymentioning

confidence: 89%

See 1 more Smart Citation

Entropy methods for CMB analysis of anisotropy and non-Gaussianity

2019

View full text Add to dashboard Cite

show abstract

“…Some partial results were found in the meantime: Coherent states were shown to be a shallow local minimum in [11], they were shown to be unique minimizers for spin 1 and 3/2 as well as for all integer Rényi entropies for all spin in [23], for spin 1 this was also shown independently in [26], sharp high spin asymptotics were settled in [3]. The conjecture was finally settled by Lieb and Solovej in [16] and further generalized in [17] and [13]. The uniqueness of the minimizers for spin greater than 3/2 is however still open.…”

Section: Wehrl Entropymentioning

confidence: 70%

Wehrl entropy, coherent states and quantum channels

Schupp¹

2022

Preprint

View full text Add to dashboard Cite

We review Wehrl's definition of a semiclassical entropy in terms of coherent states and give an introductory overview of Lieb's conjecture, its proof (including earlier results), generalizations, and the role of covariant quantum channels in this context. These structures motivate an alternative definition of coherent states and have interesting physical applications and implications.Dedicated to Elliott Lieb on the occasion of his 90th birthday.

show abstract

“…E. Lieb proved [5,25] that the Wehrl entropy is minimized by the Glauber coherent states [1,15,22,23,30], with a proof based on some difficult theorems in Fourier analysis. This result has then been generalized to symmetric SU (N ) coherent states [26,27]. It has also been proven [14,19,26] that the Husimi Q representation of coherent states majorizes the Husimi Q representation of any other quantum state, i.e.…”

mentioning

confidence: 90%

The Wehrl entropy has Gaussian optimizers

Palma

2017

Lett Math Phys

View full text Add to dashboard Cite

We determine the minimum Wehrl entropy among the quantum states with a given von Neumann entropy, and prove that it is achieved by thermal Gaussian states. This result determines the relation between the von Neumann and the Wehrl entropies. The key idea is proving that the quantumclassical channel that associates to a quantum state its Husimi Q representation is asymptotically equivalent to the Gaussian quantum-limited amplifier with infinite amplification parameter. This equivalence also permits to determine the p → q norms of the aforementioned quantum-classical channel in the two particular cases of one mode and p = q, and prove that they are achieved by thermal Gaussian states. The same equivalence permits to prove that the Husimi Q representation of a one-mode passive state (i.e. a state diagonal in the Fock basis with eigenvalues decreasing as the energy increases) majorizes the Husimi Q representation of any other one-mode state with the same spectrum, i.e. it maximizes any convex functional.

show abstract

Proof of the Wehrl-type Entropy Conjecture for Symmetric $${SU(N)}$$ Coherent States

Cited by 12 publications

References 22 publications

Entropy methods for CMB analysis of anisotropy and non-Gaussianity

Entropy methods for CMB analysis of anisotropy and non-Gaussianity

Wehrl entropy, coherent states and quantum channels

The Wehrl entropy has Gaussian optimizers

Contact Info

Product

Resources

About