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

Élliott H. Lieb, Jan Philip Solovej

arXiv:1506.07633Communications in Mathematical PhysicsMar 26, 201619 citations

Wehrl-Lieb's inequality for entropy and the uncertainty relation

Marian Grabowski

Reports on Mathematical PhysicsOct 1, 198424 citations

Proof of an Entropy Conjecture of Wehrl

Élliott H. Lieb

Springer eBooksJan 1, 200217 citations

Proof of an entropy conjecture of Wehrl

Élliott H. Lieb

Communications in Mathematical PhysicsAug 1, 1978336 citations

Maximization of the Wehrl Entropy in Finite Dimensions

Anna Bæcklund, Ingemar Bengtsson

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.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.

Jan 1, 20133 citations

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

Élliott H. Lieb, Jan Philip Solovej

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.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.

arXiv:1208.3632Acta MathematicaJan 1, 201461 citations

Proof of the Wehrl-type Entropy Conjecture for Symmetric

{SU(N)}

Coherent States