Elias Theil

Many quantum information tasks use inputs of the form $\rho^{\otimes m}$, which naturally induce permutation and unitary symmetries. We classify all channels that respect both symmetries—unitary-equivariant and permutation-invariant maps from $(\mathbb{C}^{d})^{\otimes m}$ to $(\mathbb{C}^{d})^{\otimes n}$— via their extremal points. Operationally, each extremal channel factors as \emph{unitary Schur sampling} $\rightarrow$ an \emph{irrep-level unitary-equivariant channel} $\rightarrow$ the \emph{adjoint unitary Schur sampling}. We give a streaming implementation ansatz that uses an efficient streaming implementation of unitary Schur sampling together with a resource-state primitive, and we apply it to state symmetrization, symmetric cloning, and purity amplification. In these applications we obtain polynomial-time algorithms with exponential memory improvements in $m,n$. Further, for symmetric cloning we present, to our knowledge, the first efficient (polynomial-time) algorithm with explicit memory and gate bounds.