Expanding Hermitean Operators in a Basis of Projectors on Coherent Spin States

The expectation values of a hermitean operator  in (2s+1) specific coherent states of a spin are known to determine the operator unambiguously. As shown here, (almost) any other (2s + 1) coherent states also provide a basis for selfadjoint operators. This is proven by considering the determinant of the Gram matrix associated with the coherent state projectors as a Hamiltonian of a fictitious classical spin system. State reconstruction [1] aims at parametrizing the density matrix ρ̂ of a quantum system by the expectations of appropriately chosen observables, the quorum. For a spin s, the (unnormalized) density matrix has Ns = (2s + 1) 2 independent real parameters; in [2], a particularly simple and non-redundant quorum consisting of precisely Ns projectors on coherent spin states |n〉, satisfying n · Ŝ|n〉 = h̄s|n〉, has been identified. Indeed, the density matrix ρ̂ of a spin s is determined unambiguously if one performs appropriate measurements with a traditional Stern-Gerlach apparatus. Distribute Ns axes nn, n = 1, . . . , Ns, over (2s + 1) cones about the z axis with different opening angles in such a way that the set of the (2s + 1) directions on each cone is invariant under a rotation about z by an angle 2π/(2s+ 1). Then, an (unnormalized) statistical operator ρ̂ is fixed by measuring the (2s + 1) relative frequencies ps(nn) = 〈nn|ρ̂|nn〉, that is, by the expectation values of the statistical operator ρ̂ in the coherent states |nn〉. In other words, a hermitean operator  ∈ As (which is the space of linear operators acting in the Hilbert space Hs of the spin) is fixed by the values of its Q-symbol, QA(n) = Tr[Â|n〉〈n|] = 〈n|Â|n〉 at Ns appropriately chosen points. For brevity, let us denote a set of Ns points (as well as the associated family of Ns unit vectors nn) as a ‘constellation’ N or a ‘hedgehog’ N with unit spikes nn. Independent reconstruction schemes for spin s do exist [3, 4].