Finiteness criterion for the invariant algebra of system plus quantum reference frame

Determine a necessary and sufficient condition, ideally expressed in terms of the spectral multiplicity function m_{U_R} of the compactly stabilised quantum reference frame (U_R, E, H_R), that characterizes precisely when the invariant von Neumann algebra (M_S ⊗ B(H_R))^{Ad(U_S ⊗ U_R)} is finite for a general symmetry group G = ℝ × H with H compact and a faithful normal KMS state on M_S at inverse temperature β. In particular, ascertain whether the integrability condition ∫_ℝ e^{−β ξ} m_{U_R}(ξ) dξ < ∞ is also necessary, or identify the exact criterion that holds beyond the sufficiency established in Theorem \ref{thm:type_change}.

Background

The paper proves that the invariant algebra of system and quantum reference frame observables, (M_S ⊗ B(H_R))^{Ad(U_S ⊗ U_R)}, is semifinite under natural assumptions, and finite under a sufficient condition involving the spectral multiplicity function m_{U_R} of the quantum reference frame: ∫ℝ e^{−β ξ} m{U_R}(ξ) dξ < ∞. This condition has a physical interpretation in terms of the thermal properties of the quantum reference frame and is analogous to nuclearity conditions in quantum field theory.

However, the authors only establish sufficiency of this condition in the general setting. They note that, outside a special case (trivial H and M_S a type III₁ factor) where necessity holds, the algebra need not be a factor and multiple traces may exist, leaving open whether the given integrability condition is necessary or whether a sharper, necessary-and-sufficient criterion can be formulated.