Renormalization flows for 1D mixed states and a quantum Goursat lemma

Renormalization provides a framework for relating microscopic models of physical systems to effective descriptions at larger length scales.

This procedure is studied for the boundary states of non-chiral two-dimensional topologically ordered models.

The initial data consist of renormalization fixed points built from representations of finite-dimensional $C^*$-Hopf algebras, which are then perturbed by uniform on-site noise quantum channels and repeatedly coarse-grained.

The resulting flows admit an intrinsic algebraic description in terms of completely positive maps on the $C^*$-Hopf algebra or, equivalently, positive linear functionals on its enveloping $C^*$-Hopf algebra.

Their iteration is governed by convolution powers, and convergent trajectories yield new matrix product density operator fixed points, described by finite $*$-quantum hypergroups.

This provides a concrete physical interpretation of such structures.

For finite group algebras and their duals, we provide explicit classifications via Goursat's lemma for groups.

Finally, we formulate and prove a quantum generalization of Goursat's lemma for finite-dimensional $C^*$-Hopf algebras, a result of independent interest, which gives an explicit structural description of all convergent renormalization trajectories.