Limit formulas for the trace of the functional calculus of quantum channels for \(\mathrm{SU}(2)\) (Q6623493)

This paper extends foundational results by \textit{E. H. Lieb} and \textit{J. P. Solovej} [Acta Math. 212, No. 2, 379--398 (2014; Zbl 1298.81116)], providing new limit formulas for traces in the functional calculus of quantum channels associated with \(\mathrm{SU}(2)\) representations. The author broadens the scope of quantum channels to include all components of \(\mathrm{SU}(2)\) tensor decompositions, not only the leading component and establishes the trace limiting formula for \(\phi \in C([0,1])\):\N\[\N\lim_{\nu \to \infty} \frac{1}{\nu} \operatorname{Tr}(\phi(\mathcal{T}_{\mu,k}^\nu(R_\mu^*(f)))) = \int_{\mathbb C} \phi(E_{\mu,k}(f)) \frac{dz}{\pi(1 + |z|^2)^2},\N\]\Nfor any \(f \in C(\mathbb{CP}^1)\) with \(\int_{\mathbb{CP}^1} f(z) \frac{dz}{\pi(1 + |z|^2)^2} = 1\) and Toeplitz operator \(R_\mu^*(f) \geq 0\), where \( E_{\mu, k}(f) \) incorporates the Berezin transforms. Some additional related results are also presented.