Wehrl's entropy of spin states and Lieb's conjecture
C T Lee · Journal of Physics A Mathematical and General · 1988
Wehrl's entropy is the entropy of the probability distribution in the phase space corresponding to the Q representation (antinormal ordering of operators) of a quantum state in terms of coherent states. The Hilbert space of a system of N spins (or N two-level atoms) is of 2 N dimensions. The subspace with maximum total spin N/2 is of N+1 dimensions. All discussions are restricted within this subspace. The probability amplitude for an arbitrary pure state in this subspace is a polynomial of degree N which can be factorised into the product of N linear factors and each root can be identified with one point on the unit sphere. Hence, an arbitrary state in the subspace can be represented geometrically by N unit vectors. The general expression for the Wehrl entropy is obtained as a finite series expansion in terms of some symmetric functions of sin 2 ( omega ij /2), where omega ij is the angle between a pair of unit vectors. The special cases of coherent and almost coherent spin states are then considered.