Lueders Theorem for Coherent-State POVMs

Lueders theorem states that two observables commute if measuring one of them does not disturb the measurement outcomes of the other. We study measurements which are described by continuous positive operator-valued measurements (or POVMs) associated with coherent states on a Lie group. In general, operators turn out to be invariant under the Lueders map if their P- and Q-symbols coincide. For a spin corresponding to SU(2), the identity is shown to be the only operator with this property. For a particle, a countable family of linearly independent operators is identified which are invariant under the Lueders map generated by the coherent states of the Heisenberg-Weyl group, H_3. The Lueders map is also shown to implement the anti-normal ordering of creation and annihilation operators of a particle.